WEBVTT 1 00:00:06.439 --> 00:00:09.994 Game Basics Fundamental Mathematics for Games 3: Trigonometry and Linearity 2 00:00:27.294 --> 00:00:28.916 Hello, everyone 3 00:00:28.916 --> 00:00:30.839 This is Lee Deuk-woo from Game Mathematics 4 00:00:30.839 --> 00:00:34.910 Today, we will learn about trigonometry and linearity 5 00:00:34.910 --> 00:00:39.120 The term trigonometry might sound intimidating 6 00:00:39.120 --> 00:00:42.488 We will focus on the essential trigonometric concepts 7 00:00:42.488 --> 00:00:45.560 used in game development 8 00:00:45.560 --> 00:00:48.160 We will also explore what linearity is 9 00:00:48.160 --> 00:00:53.000 and expand it to discuss linear mappings and transformations 10 00:00:53.480 --> 00:00:57.074 Trigonometry 11 00:00:57.520 --> 00:01:01.430 Let’s first summarize the concept of trigonometry 12 00:01:01.430 --> 00:01:04.400 What exactly is trigonometry? 13 00:01:04.400 --> 00:01:07.979 First, the word "trigonometry" 14 00:01:07.979 --> 00:01:13.440 fundamentally deals with triangles 15 00:01:13.440 --> 00:01:16.800 Specifically, it refers to right-angled triangles 16 00:01:16.800 --> 00:01:19.293 A right-angled triangle is 17 00:01:19.293 --> 00:01:24.279 a triangle with one angle measuring exactly 90 degrees 18 00:01:24.279 --> 00:01:28.919 A right-angled triangle is composed of three sides 19 00:01:28.919 --> 00:01:29.791 The hypotenuse 20 00:01:29.791 --> 00:01:32.718 the longest side opposite the right angle 21 00:01:32.718 --> 00:01:35.040 the adjacent side (base) 22 00:01:35.040 --> 00:01:38.931 and the opposite side (height) 23 00:01:38.931 --> 00:01:44.980 These three sides define a right-angled triangle 24 00:01:44.980 --> 00:01:50.350 We classify these as the elements of a right-angled triangle 25 00:01:50.350 --> 00:01:55.262 From the three sides of a right-angled triangle 26 00:01:55.262 --> 00:01:58.599 we define the ratio of two sides as trigonometric ratios 27 00:01:58.599 --> 00:02:02.279 There are six possible ratios in total 28 00:02:02.279 --> 00:02:06.219 However, three of these are most commonly used 29 00:02:06.219 --> 00:02:08.204 The first is sin 30 00:02:08.204 --> 00:02:11.279 It is the ratio of the opposite side to the hypotenuse 31 00:02:11.279 --> 00:02:12.367 The second is cos 32 00:02:12.367 --> 00:02:13.964 It is the ratio of the adjacent side to the hypotenuse 33 00:02:13.964 --> 00:02:15.860 The third is tan 34 00:02:15.860 --> 00:02:17.000 It is the ratio of the opposite side to the adjacent side 35 00:02:17.000 --> 00:02:20.461 These three ratios are primarily 36 00:02:20.461 --> 00:02:24.160 used in surveying and related fields 37 00:02:24.160 --> 00:02:26.510 Hence, they are the most frequently applied 38 00:02:26.510 --> 00:02:30.895 As we progress further 39 00:02:30.895 --> 00:02:34.360 although trigonometric functions are widely used across many fields 40 00:02:34.360 --> 00:02:38.779 in graphics, the sin and cos functions 41 00:02:38.779 --> 00:02:40.960 are the most commonly utilized 42 00:02:40.960 --> 00:02:43.727 As for the tan function 43 00:02:43.727 --> 00:02:47.880 it will be explained later in the context of inverse functions 44 00:02:47.880 --> 00:02:51.579 For now, don’t try to learn everything about trigonometric functions 45 00:02:51.579 --> 00:02:53.357 Focus first on understanding sin and cos together 46 00:02:53.357 --> 00:02:56.719 and think of them as a pair to learn initially 47 00:02:56.719 --> 00:03:00.630 It’s helpful to grasp the concept of the tan function 48 00:03:00.630 --> 00:03:03.086 but in practice, when using tan 49 00:03:03.086 --> 00:03:08.037 we often rely on its inverse function 50 00:03:08.037 --> 00:03:13.281 to calculate angles associated with vectors 51 00:03:13.281 --> 00:03:17.969 Thus, our primary focus will be on these three applications 52 00:03:17.969 --> 00:03:22.370 We’ll simplify and organize the content accordingly 53 00:03:22.370 --> 00:03:29.193 Using any two sides of a right-angled triangle 54 00:03:29.193 --> 00:03:32.619 we calculate the ratio, which is called a trigonometric ratio 55 00:03:32.619 --> 00:03:37.299 We extend this trigonometric ratio into the concept of a function 56 00:03:37.299 --> 00:03:42.738 This is based on the angle of the right-angled triangle 57 00:03:42.738 --> 00:03:44.775 used to form the hypotenuse 58 00:03:44.775 --> 00:03:50.327 where the angle between the base and hypotenuse becomes the domain 59 00:03:50.327 --> 00:03:54.846 When we take the value of this ratio 60 00:03:54.846 --> 00:03:59.171 the value is always between -1 and 1 61 00:03:59.171 --> 00:04:00.855 Now, if we represent this in coordinates 62 00:04:00.855 --> 00:04:03.528 in the Cartesian coordinate system 63 00:04:03.528 --> 00:04:07.481 the value of the ratio combines the positive and negative regions 64 00:04:07.481 --> 00:04:14.080 and is confined to a specific range between -1 and 1 65 00:04:14.080 --> 00:04:15.880 Thus, the angle we take as the domain 66 00:04:15.880 --> 00:04:19.720 is set as the domain 67 00:04:19.720 --> 00:04:22.016 For the resulting values 68 00:04:22.016 --> 00:04:25.480 we set the range between -1 and 1 69 00:04:25.480 --> 00:04:29.160 This establishes a correspondence 70 00:04:29.160 --> 00:04:32.279 This can be seen as a correspondence between two sets 71 00:04:32.279 --> 00:04:34.729 Here, the domain corresponds to 72 00:04:34.729 --> 00:04:37.160 all real numbers 73 00:04:37.160 --> 00:04:39.620 Meanwhile, the codomain and range 74 00:04:39.620 --> 00:04:44.050 are restricted to the range of -1 to 1 75 00:04:44.050 --> 00:04:46.777 Thus, the domain element x 76 00:04:46.777 --> 00:04:50.880 and the codomain element y 77 00:04:50.880 --> 00:04:54.713 are related to the trigonometric ratio we discussed earlier 78 00:04:54.713 --> 00:04:59.040 By substituting a value into the domain 79 00:04:59.040 --> 00:05:02.399 we can express it as the function y = sinx 80 00:05:02.399 --> 00:05:06.600 This is what we call a trigonometric function 81 00:05:06.600 --> 00:05:08.350 In a way, trigonometric functions 82 00:05:08.350 --> 00:05:14.250 generalize trigonometric ratios in the Cartesian coordinate system 83 00:05:14.250 --> 00:05:21.529 covering all angles in the 360 degree plane 84 00:05:21.529 --> 00:05:23.588 This generalized correspondence 85 00:05:23.588 --> 00:05:27.519 is what we call trigonometric functions 86 00:05:27.519 --> 00:05:30.524 When we use trigonometric functions 87 00:05:30.524 --> 00:05:34.816 we don't limit them to right-angled triangles 88 00:05:34.816 --> 00:05:37.449 but apply them to the entire plane 89 00:05:37.449 --> 00:05:42.760 analyzing them based on the unit circle 90 00:05:42.760 --> 00:05:47.320 A unit circle refers to a circle with a radius of 1 91 00:05:47.320 --> 00:05:49.676 It specifically means a circle with a radius of 1 92 00:05:49.676 --> 00:05:53.219 Now, consider a right-angled triangle 93 00:05:53.219 --> 00:05:55.308 inscribed in this circle with a radius of 1 94 00:05:55.308 --> 00:05:59.660 In such a case, when we look at the right-angled triangle 95 00:05:59.660 --> 00:06:02.341 the hypotenuse 96 00:06:02.341 --> 00:06:05.900 always corresponds to a point 97 00:06:05.900 --> 00:06:09.368 on the arc of the unit circle 98 00:06:09.368 --> 00:06:14.930 Thus, the value of the hypotenuse is always 1, the radius of the circle 99 00:06:14.930 --> 00:06:19.528 In this setup, when we define trigonometric functions 100 00:06:19.528 --> 00:06:24.060 the base divided by the hypotenuse is cos, and the height divided by the hypotenuse is sin 101 00:06:24.060 --> 00:06:28.027 So, as we continuously rotate this hypotenuse vector 102 00:06:28.027 --> 00:06:31.767 around one or two full rotations 103 00:06:31.767 --> 00:06:34.839 we obtain the values of these curves 104 00:06:34.839 --> 00:06:41.600 These are the graphs of the sin and cosine functions 105 00:06:41.600 --> 00:06:46.071 These are well-known visuals, so you search online 106 00:06:46.071 --> 00:06:48.800 you will be able to find theme easily 107 00:06:48.800 --> 00:06:51.795 The values of these trigonometric ratios 108 00:06:51.795 --> 00:06:55.765 show how they change with respect to the angle 109 00:06:55.765 --> 00:06:57.780 We can observe this 110 00:06:57.780 --> 00:07:00.210 While it resets after one full rotation 111 00:07:00.210 --> 00:07:02.589 it essentially continues indefinitely 112 00:07:02.589 --> 00:07:04.959 We can also consider rotations in the opposite direction 113 00:07:04.959 --> 00:07:10.368 This means that all real-number values 114 00:07:10.368 --> 00:07:13.719 can be used for rotation 115 00:07:13.719 --> 00:07:16.263 To summarize these graphs 116 00:07:16.263 --> 00:07:19.679 the sin and cos functions can be expressed as follows 117 00:07:19.679 --> 00:07:23.204 First, as previously mentioned, sin and cos 118 00:07:23.204 --> 00:07:28.312 always arewithin a range of -1 to 1 119 00:07:28.312 --> 00:07:31.099 which is the consistent, repeating pattern they follow 120 00:07:31.099 --> 00:07:35.358 Both functions share the same pattern 121 00:07:35.358 --> 00:07:37.335 Another shared characteristic is 122 00:07:37.335 --> 00:07:42.639 that their repeating patterns complete one full cycle 123 00:07:42.639 --> 00:07:45.134 commonly referred to as 360 degrees 124 00:07:45.134 --> 00:07:48.679 and then repeat 125 00:07:48.679 --> 00:07:52.207 This repetition over 360 degrees 126 00:07:52.207 --> 00:07:55.338 is also referred to as 2 pi radian 127 00:07:55.338 --> 00:07:59.027 Radians and degrees 128 00:07:59.027 --> 00:08:00.450 about angular measurement systems 129 00:08:00.450 --> 00:08:02.421 will be explained later 130 00:08:02.421 --> 00:08:04.469 Here, as a supplementary explanation 131 00:08:15.875 --> 00:08:18.138 let’s clarify the 360 degree notation 132 00:08:18.138 --> 00:08:21.480 Next, this is an important point 133 00:08:21.480 --> 00:08:23.490 A characteristic of trigonometric functions is 134 00:08:23.490 --> 00:08:27.569 imagine folding them along the y-axis 135 00:08:27.569 --> 00:08:31.079 If you fold the left and right sides together 136 00:08:31.079 --> 00:08:35.787 the left part of the sin function 137 00:08:35.787 --> 00:08:40.179 always has the opposite sign of the right part 138 00:08:40.179 --> 00:08:43.091 When mirrored this way 139 00:08:43.091 --> 00:08:46.429 the two parts become vertically symmetric 140 00:08:46.429 --> 00:08:47.519 The same applies to the other side 141 00:08:47.519 --> 00:08:50.746 When folded, this side shifts over here 142 00:08:50.746 --> 00:08:53.313 The values on the left 143 00:08:53.313 --> 00:08:57.280 always take the opposite direction of the values on the right 144 00:08:57.280 --> 00:09:00.895 Thus, while the values differ when folding the sine function 145 00:09:00.895 --> 00:09:03.380 it exhibits a specific property 146 00:09:03.380 --> 00:09:04.592 of being symmetric 147 00:09:04.592 --> 00:09:07.927 about the x-axis 148 00:09:07.927 --> 00:09:10.240 This can be described as its symmetry property 149 00:09:10.240 --> 00:09:15.050 In contrast, the cos function is symmetric about the y axis 150 00:09:15.050 --> 00:09:17.763 When folded in this manner 151 00:09:17.763 --> 00:09:23.060 it perfectly mirrors itself, like a decalcomania 152 00:09:23.060 --> 00:09:24.956 Therefore, for the cosine function 153 00:09:24.956 --> 00:09:32.169 it is always symmetric with respect to negative inputs 154 00:09:32.169 --> 00:09:35.080 which means it retains the same value 155 00:09:35.080 --> 00:09:38.862 These two properties are key characteristics of sin and cos functions 156 00:09:38.862 --> 00:09:43.130 and understanding them will be very useful later 157 00:09:43.130 --> 00:09:46.417 Next, let's discuss the tan function 158 00:09:46.417 --> 00:09:48.821 For the tan function 159 00:09:48.821 --> 00:09:54.011 it does not exist at cos 90 degrees and cos 270 degrees 160 00:09:54.011 --> 00:09:56.120 There are no solutions at these points 161 00:09:56.120 --> 00:09:58.466 It has this peculiar characteristic 162 00:09:58.466 --> 00:10:02.100 The same applies to -90 degrees and -270 degrees 163 00:10:02.100 --> 00:10:06.823 Let’s explore why it doesn’t exist 164 00:10:06.823 --> 00:10:11.664 First, consider the sin and cos functions 165 00:10:11.664 --> 00:10:14.842 When we think about the unit circle 166 00:10:14.842 --> 00:10:23.605 let’s look at the values generated by sin and cos in the unit circle 167 00:10:23.605 --> 00:10:25.261 Here, the base 168 00:10:25.261 --> 00:10:28.504 and the hypotenuse is such that the hypotenuse is always 1 169 00:10:28.504 --> 00:10:34.298 so the base value is naturally cos theta 170 00:10:34.298 --> 00:10:36.862 This base corresponds to the x value 171 00:10:36.862 --> 00:10:39.452 Next, the height 172 00:10:39.452 --> 00:10:44.159 is the height divided by the base, which is sin 173 00:10:44.159 --> 00:10:47.609 So, for any arbitrary 174 00:10:47.609 --> 00:10:53.615 angle theta, if we draw a unit circle 175 00:10:53.615 --> 00:10:55.627 the region with a radius of 1 176 00:10:55.627 --> 00:10:58.130 is affected by theta 177 00:10:58.130 --> 00:10:59.699 Its coordinates are always 178 00:10:59.699 --> 00:11:03.060 cos theta, sin theta 179 00:11:03.060 --> 00:11:05.080 Then, what about the height divided by the base? 180 00:11:05.080 --> 00:11:08.496 This is the tan function, and the height divided by the base 181 00:11:08.496 --> 00:11:11.060 has the base value cos as x 182 00:11:11.060 --> 00:11:12.966 and the height value as sin 183 00:11:12.966 --> 00:11:16.040 So, it is sin theta / cos theta 184 00:11:16.040 --> 00:11:21.466 However, the denominator, which is the base, corresponds to the cos function 185 00:11:21.466 --> 00:11:24.436 A denominator of zero cannot exist 186 00:11:24.436 --> 00:11:28.301 Therefore, at points where cos x = 0 187 00:11:28.301 --> 00:11:31.436 such as 90 and 270 degrees 188 00:11:31.436 --> 00:11:35.984 the tan function naturally doesn't exist 189 00:11:35.984 --> 00:11:42.318 Thus, as the intervals of 90 and 270 degrees 190 00:11:42.318 --> 00:11:46.030 it exhibits a pattern of having no solutions 191 00:11:46.030 --> 00:11:50.189 Next, there is another crucial formula to discuss 192 00:11:50.189 --> 00:11:55.120 In right-angled triangles, we have the Pythagorean theorem 193 00:11:55.120 --> 00:11:58.298 Previously, we used it to find the shortest distance from the origin 194 00:11:58.298 --> 00:12:01.110 Using the Pythagorean theorem 195 00:12:01.110 --> 00:12:06.887 we can deduce that the square of the base plus the square of the height 196 00:12:06.887 --> 00:12:11.971 equals the square of the hypotenuse 197 00:12:11.971 --> 00:12:15.303 Since the hypotenuse is always fixed at 1 198 00:12:15.303 --> 00:12:17.694 the square of the height results in 1 199 00:12:17.694 --> 00:12:21.245 and with the base as cos and the height as sin 200 00:12:21.245 --> 00:12:25.446 we arrive at cos^2 theta + sin^2 theta = 1 201 00:12:25.446 --> 00:12:30.654 This relationship is always satisfied 202 00:12:30.654 --> 00:12:33.644 Such an equation always holds true 203 00:12:33.644 --> 00:12:36.498 You might ask, is this because it’s a unit circle? 204 00:12:36.498 --> 00:12:38.506 Let’s consider the question 205 00:12:38.506 --> 00:12:44.852 If we set the radius of a circle to an arbitrary value r 206 00:12:44.852 --> 00:12:48.239 and if the radius of the arc is r 207 00:12:48.239 --> 00:12:50.055 with r being any arbitrary value 208 00:12:50.055 --> 00:12:55.674 this is essentially scaling the vector of magnitude 1 209 00:12:55.674 --> 00:12:59.567 by r 210 00:12:59.567 --> 00:13:02.817 It’s a scalar-based scaling of the vector 211 00:13:02.817 --> 00:13:06.337 This can be viewed as scalar multiplication of a vector 212 00:13:06.337 --> 00:13:11.678 Therefore, the vector with magnitude 1 always has 213 00:13:11.678 --> 00:13:13.625 values of cos theta, sin theta 214 00:13:13.625 --> 00:13:16.634 scaling it by r gives the scaled values 215 00:13:16.634 --> 00:13:20.701 Based on this equation, the x-value becomes 216 00:13:20.701 --> 00:13:24.397 r cos theta, and the y value becomes r sin theta 217 00:13:24.397 --> 00:13:26.697 Now, in such a right-angled triangle 218 00:13:26.697 --> 00:13:32.021 let's check if cos^2 theta + sin^2 theta = 1 still holds 219 00:13:32.021 --> 00:13:34.946 Since it is based on trigonometric ratios 220 00:13:34.946 --> 00:13:37.996 the base is r^2 cos ^2 theta 221 00:13:37.996 --> 00:13:41.694 the height is r^2sin^2 theta 222 00:13:41.694 --> 00:13:44.238 and the hypotenuse is r ^2 223 00:13:44.238 --> 00:13:46.683 Dividing through by r^2, we get 224 00:13:46.683 --> 00:13:50.258 cos^2 theta + sin^2 theta = 1 225 00:13:50.258 --> 00:13:52.965 which regardless of the radius 226 00:13:52.965 --> 00:13:57.545 we can understand that this formula is universally valid 227 00:13:57.545 --> 00:13:59.469 As previously explained 228 00:13:59.469 --> 00:14:05.036 for a radius r, the x value is r cos theta 229 00:14:05.036 --> 00:14:08.417 and the y value is r sin theta 230 00:14:08.417 --> 00:14:11.984 allowing us to decompose any point on the circle 231 00:14:11.984 --> 00:14:17.872 into x and y components 232 00:14:17.872 --> 00:14:21.367 One more useful point to note 233 00:14:21.367 --> 00:14:24.455 In the real vector space 234 00:14:24.455 --> 00:14:29.397 R^2, the standard basis vectors were (1,0) and (0,1) 235 00:14:29.397 --> 00:14:33.328 These are also vectors with a magnitude of 1 236 00:14:33.328 --> 00:14:36.312 If these vectors have a magnitude of 1 237 00:14:36.312 --> 00:14:39.793 as mentioned earlier, they can be generalized 238 00:14:39.793 --> 00:14:42.750 to represent all vectors of magnitude 1 in the plane 239 00:14:42.750 --> 00:14:47.486 as cos theta, sin theta 240 00:14:47.486 --> 00:14:53.080 Two specific examples are (1,0) and (0,1) 241 00:14:53.080 --> 00:14:54.595 These correspond 242 00:14:54.595 --> 00:14:59.199 to angles of 0 and 90 degrees 243 00:14:59.199 --> 00:15:03.723 Thus, the first standard basis vector e1 244 00:15:03.723 --> 00:15:06.720 is cos 0 degrees, sin 0 degrees 245 00:15:06.720 --> 00:15:09.585 with values (1,0) 246 00:15:09.585 --> 00:15:13.092 The second, e2, is cos 90 degrees, sin 90 degrees 247 00:15:13.092 --> 00:15:18.684 corresponding to (0,1) 248 00:15:18.684 --> 00:15:22.097 In this way, we’ve examined the various properties 249 00:15:22.097 --> 00:15:24.139 of the sin and cos functions 250 00:15:24.139 --> 00:15:27.416 Always think of the unit circle with a radius of 1 251 00:15:27.416 --> 00:15:30.548 and the right-angled triangles formed within that unit circle 252 00:15:30.548 --> 00:15:32.387 If you always associate them together 253 00:15:32.387 --> 00:15:37.574 even if you forget the memorization of sin and cos 254 00:15:37.574 --> 00:15:40.496 it won't be too difficult to recall them 255 00:15:40.496 --> 00:15:45.120 Always remember trigonometric functions alongside the unit circle 256 00:15:45.120 --> 00:15:47.729 Now, let’s learn about the measurement of angles 257 00:15:47.729 --> 00:15:51.612 In daily life, when referring to circles 258 00:15:51.612 --> 00:15:54.753 we usually use 360 degrees to describe them 259 00:15:54.753 --> 00:15:57.968 However, mathematics doesn’t use degrees like this 260 00:15:57.968 --> 00:16:00.694 This system is called the degree system 261 00:16:00.694 --> 00:16:02.906 When we use 360 as 262 00:16:02.906 --> 00:16:05.493 a reference to develop something 263 00:16:05.493 --> 00:16:08.872 we may question, "Why divide by 360?" 264 00:16:08.872 --> 00:16:12.120 Such questions arise 265 00:16:12.120 --> 00:16:15.669 So, using the concept of a unit length of 1 266 00:16:15.669 --> 00:16:19.110 how can we measure a circle? 267 00:16:19.110 --> 00:16:22.684 We need a separate measurement method for this 268 00:16:22.684 --> 00:16:25.272 This method is called the radian system 269 00:16:25.272 --> 00:16:26.229 In the radian system 270 00:16:26.229 --> 00:16:30.080 let me briefly explain how it works 271 00:16:30.080 --> 00:16:33.095 First, always associate the radian system with 272 00:16:33.095 --> 00:16:37.154 a unit circle with a radius of 1, cut in half 273 00:16:37.154 --> 00:16:38.922 Imagine a semicircle 274 00:16:38.922 --> 00:16:41.492 Think of a semicircle with a radius of 1 275 00:16:41.492 --> 00:16:44.090 Then it will create a shape like this, right? 276 00:16:44.090 --> 00:16:47.719 Now, let’s stretch this semicircle to the right 277 00:16:47.719 --> 00:16:48.815 Stretch it by 1 unit 278 00:16:48.815 --> 00:16:53.011 The end here will move to the center 279 00:16:53.011 --> 00:16:55.694 This forms a semicircle 280 00:16:55.694 --> 00:16:58.988 Think of it as a string, and hold it here 281 00:16:58.988 --> 00:17:01.397 Now imagine pulling it straight 282 00:17:01.397 --> 00:17:06.192 This creates a straight line 283 00:17:06.192 --> 00:17:09.410 Then, compared to the original radius of 1 284 00:17:09.410 --> 00:17:11.809 we can measure 285 00:17:11.809 --> 00:17:15.861 how many times it has stretched 286 00:17:15.861 --> 00:17:18.740 The arc length can be measured this way 287 00:17:18.740 --> 00:17:21.454 This length corresponds to the famous 288 00:17:21.454 --> 00:17:25.079 number pi, starting with 3.141592 289 00:17:25.079 --> 00:17:28.653 However, as you know, it’s not an exact number 290 00:17:28.653 --> 00:17:31.079 It is the irrational number pi 291 00:17:31.079 --> 00:17:36.366 Thus, a multiple of approximately 3.14 292 00:17:36.366 --> 00:17:39.079 gives us the arc length 293 00:17:39.079 --> 00:17:41.841 So, what do we do in this case? 294 00:17:41.841 --> 00:17:44.604 We reverse the process by rewinding 295 00:17:44.604 --> 00:17:50.086 Place this line back to its original position 296 00:17:50.086 --> 00:17:52.693 Rewind it in this manner 297 00:17:52.693 --> 00:18:01.918 As we rewind, how much do we rewind? 298 00:18:01.918 --> 00:18:04.010 We rewind by a length of 1 299 00:18:04.010 --> 00:18:09.738 If we rewind an arc length of 1 300 00:18:09.738 --> 00:18:15.059 both the arc length and the radius are 1 301 00:18:15.059 --> 00:18:17.592 In such a sector 302 00:18:17.592 --> 00:18:23.059 all the sides forming the sector have a length of 1 303 00:18:23.059 --> 00:18:26.470 At this point, we can determine the angle 304 00:18:26.470 --> 00:18:30.079 Let’s define this as the standard value 305 00:18:30.079 --> 00:18:34.188 The angle corresponding to an arc length of 1 306 00:18:34.188 --> 00:18:37.821 is defined as the base angle 307 00:18:37.821 --> 00:18:40.327 This is the radian 308 00:18:40.327 --> 00:18:44.162 As you might guess, radians 309 00:18:44.162 --> 00:18:46.325 don’t result in exact values 310 00:18:46.325 --> 00:18:51.079 It corresponds to approximately 52.3 degrees, which is also an irrational number 311 00:18:51.079 --> 00:18:56.569 This angle is approximately 52.2958, repeating infinitely 312 00:18:56.569 --> 00:19:05.190 So, how much more should we connect 313 00:19:05.190 --> 00:19:08.287 to restore it to a semicircle? 314 00:19:08.287 --> 00:19:14.979 Multiplying 52.2958 by approximately 3.14 315 00:19:14.979 --> 00:19:16.955 or pi-times 316 00:19:16.955 --> 00:19:22.614 restores it to exactly 180 degrees 317 00:19:22.614 --> 00:19:26.473 Therefore, 180 degrees essentially means 318 00:19:26.473 --> 00:19:33.995 multiplying the radian value corresponding to 52.3 degrees by approximately 3.14 319 00:19:33.995 --> 00:19:36.119 It aligns closely with this 320 00:19:36.119 --> 00:19:39.010 Thus, the result is pi radians 321 00:19:39.010 --> 00:19:42.629 In the degree system we commonly use in daily life 322 00:19:42.629 --> 00:19:44.666 1 degree corresponds to 323 00:19:44.666 --> 00:19:48.385 pi/180 radians 324 00:19:48.385 --> 00:19:50.594 as derived from the formula 325 00:19:50.594 --> 00:19:54.854 1 radian, on the other hand, is the value of 180 / pi 326 00:19:54.854 --> 00:19:57.406 This is how it can be understood 327 00:19:57.406 --> 00:20:01.771 If this formula seems too complex 328 00:20:01.771 --> 00:20:03.129 just think of it this way 329 00:20:03.129 --> 00:20:04.943 even though it's an irrational number 330 00:20:04.943 --> 00:20:07.767 a radian is just over 50 degrees 331 00:20:07.767 --> 00:20:12.663 So, how many times smaller than 180 degrees is 53 degrees? 332 00:20:12.663 --> 00:20:16.502 It’s approximately 3.14 times smaller 333 00:20:16.502 --> 00:20:20.782 In other words, divide 180 by 3.14 334 00:20:20.782 --> 00:20:26.338 and you’ll roughly understand the relationship 335 00:20:26.338 --> 00:20:29.247 between radians and degrees without much confusion 336 00:20:29.247 --> 00:20:31.939 For instance 90 degrees 337 00:20:31.939 --> 00:20:33.717 is half of 180 degrees 338 00:20:33.717 --> 00:20:35.643 so it is pi/2 radians 339 00:20:35.643 --> 00:20:38.229 60 degrees is pi/3 radians 340 00:20:38.229 --> 00:20:44.188 Since these commonly used angles are expressed in radians 341 00:20:44.188 --> 00:20:47.816 frequently used values like these 342 00:20:47.816 --> 00:20:49.456 should be memorized for quick recall 343 00:20:49.456 --> 00:20:51.643 This will make it easier to use later 344 00:20:51.643 --> 00:20:54.129 So, let’s summarize this 345 00:20:54.129 --> 00:20:58.658 That’s the explanation of the radian system 346 00:20:58.658 --> 00:21:01.079 Why do we learn the radian system? 347 00:21:01.079 --> 00:21:04.921 Because it is widely used in mathematics and 348 00:21:04.921 --> 00:21:09.772 when dealing with sin, cos, and other trigonometric functions in computers 349 00:21:09.772 --> 00:21:13.792 everything is based on the radian system 350 00:21:13.792 --> 00:21:15.306 However, when learning 351 00:21:15.306 --> 00:21:17.561 to make things easier to understand 352 00:21:17.561 --> 00:21:19.742 we sometimes use the degree system as well 353 00:21:19.742 --> 00:21:22.467 While radians are the standard 354 00:21:22.467 --> 00:21:25.851 if radians are used without explicitly labeling them 355 00:21:25.851 --> 00:21:28.125 it’s understood to be based on the radian system 356 00:21:28.125 --> 00:21:30.367 This means it uses radians as the unit 357 00:21:30.367 --> 00:21:32.990 radians as the unit 358 00:21:32.990 --> 00:21:34.510 If we use degrees 359 00:21:34.510 --> 00:21:36.528 and degrees are used for better understanding 360 00:21:36.528 --> 00:21:38.049 you will see the degree symbol 361 00:21:38.049 --> 00:21:41.762 clearly marked with a circle symbol 362 00:21:41.762 --> 00:21:44.148 Please understand it this way 363 00:21:44.148 --> 00:21:48.831 We’ve now covered the definition of trigonometric functions, their applications, and formulas 364 00:21:48.831 --> 00:21:52.158 We’ve also explored the characteristics of trigonometric functions 365 00:21:52.158 --> 00:21:55.608 Additionally, we’ve reviewed the differences between degrees and radians 366 00:21:55.608 --> 00:21:59.723 Let’s look at where these concepts are commonly applied 367 00:21:59.723 --> 00:22:02.824 The most common application now is 368 00:22:02.824 --> 00:22:07.018 when rotating an object by an angle theta 369 00:22:07.018 --> 00:22:10.178 Trigonometric functions are frequently used for this 370 00:22:10.178 --> 00:22:14.816 So, what does it mean to rotate an object by theta 371 00:22:14.816 --> 00:22:17.723 To understand how this precess works 372 00:22:17.723 --> 00:22:20.606 if you understand this mechanism 373 00:22:20.606 --> 00:22:25.574 it will be very useful later when learning about matrices 374 00:22:25.574 --> 00:22:29.029 and linear transformations 375 00:22:29.029 --> 00:22:33.975 First, before the transformation 376 00:22:33.975 --> 00:22:37.490 before the object changes or rotates 377 00:22:37.490 --> 00:22:44.366 let’s assume an arbitrary element in the real vector space R2 is x,y 378 00:22:44.366 --> 00:22:50.018 this x,y can be expressed as a linear combination 379 00:22:50.018 --> 00:22:53.583 like this 380 00:22:53.583 --> 00:22:56.802 and view it this way 381 00:22:56.802 --> 00:23:02.977 At this point, if the space undergoes force 382 00:23:02.977 --> 00:23:05.831 or any external factor causing transformation 383 00:23:05.831 --> 00:23:10.356 then it’s not just a part of the space that transforms 384 00:23:10.356 --> 00:23:12.901 but the entire space changes 385 00:23:12.901 --> 00:23:16.422 However, since vector spaces are infinite 386 00:23:16.422 --> 00:23:19.212 we cannot individually manipulate every single vector 387 00:23:19.212 --> 00:23:23.426 in the infinite space for transformation 388 00:23:23.426 --> 00:23:25.716 So how do we approach this concept? 389 00:23:25.716 --> 00:23:29.659 Ultimately, all elements are constructed 390 00:23:29.659 --> 00:23:34.334 from two standard basis vectors 391 00:23:34.334 --> 00:23:36.218 that form the vector space 392 00:23:36.218 --> 00:23:38.633 If these basis vectors change 393 00:23:38.633 --> 00:23:43.967 it’s like altering the foundation of a house, changing its structure 394 00:23:43.967 --> 00:23:48.831 By changing the central pillars of the vector space 395 00:23:48.831 --> 00:23:52.243 you don’t have to trace every infinite element individually 396 00:23:52.243 --> 00:23:56.237 but you can infer how the overall structure changes 397 00:23:56.237 --> 00:23:58.004 by imagining it in your mind 398 00:23:58.004 --> 00:24:01.079 This allows you to visualize the transformation 399 00:24:01.079 --> 00:24:05.787 By tracking changes in the standard basis vectors 400 00:24:05.787 --> 00:24:09.788 and replacing the original basis vectors 401 00:24:09.788 --> 00:24:11.980 with two new basis vectors 402 00:24:11.980 --> 00:24:14.708 we multiply x and y by each 403 00:24:14.708 --> 00:24:19.574 and represent them as a relationship with new vectors 404 00:24:19.574 --> 00:24:26.277 This allows us to understand how the space transforms 405 00:24:26.277 --> 00:24:29.375 Rotation here means 406 00:24:29.375 --> 00:24:32.448 that the new two basis vectors have a magnitude of 1 407 00:24:32.448 --> 00:24:35.564 just like the original e1 and e2 408 00:24:35.564 --> 00:24:38.517 They maintain orthogonality 409 00:24:38.517 --> 00:24:41.792 with e1 and e2 being 90 degrees apart 410 00:24:41.792 --> 00:24:45.996 Even their orientation 411 00:24:45.996 --> 00:24:48.002 is preserved during transformation 412 00:24:48.002 --> 00:24:50.930 This is called a rotational transformation 413 00:24:50.930 --> 00:24:54.109 I’ll explain this in detail later 414 00:24:54.109 --> 00:25:00.148 So, (1,0) rotates by an angle theta 415 00:25:00.148 --> 00:25:04.257 and (0,1) also rotates by theta 416 00:25:04.257 --> 00:25:10.123 In the diagram, the standard basis vector on the x-axis 417 00:25:10.123 --> 00:25:12.949 e1 rotates by theta 418 00:25:12.949 --> 00:25:17.713 and its coordinates become cos theta, sin theta 419 00:25:17.713 --> 00:25:22.403 Then, when (0,1) rotates by theta 420 00:25:22.403 --> 00:25:23.842 if we draw a right triangle here 421 00:25:23.842 --> 00:25:27.158 the two triangles are similar 422 00:25:27.158 --> 00:25:32.509 The height here moves to this height 423 00:25:32.509 --> 00:25:33.990 However, it becomes the x-value 424 00:25:33.990 --> 00:25:36.841 Here, the y-value becomes the x-value 425 00:25:36.841 --> 00:25:40.366 and the x-value flips to the y-value 426 00:25:40.366 --> 00:25:44.498 As you can see, it was positive here 427 00:25:44.498 --> 00:25:47.079 but the x-value becomes negative here 428 00:25:47.079 --> 00:25:49.297 x and y swap places 429 00:25:49.297 --> 00:25:53.197 and the x value takes on 430 00:25:53.197 --> 00:25:57.742 a negative value in the opposite direction 431 00:25:57.742 --> 00:26:02.843 So, when (0,1) rotates by theta 432 00:26:02.843 --> 00:26:08.683 its values become -sin theta, cos theta 433 00:26:08.683 --> 00:26:15.665 Ultimately, this means the standard basis vector e1 becomes cos theta, sin theta 434 00:26:15.665 --> 00:26:21.228 and e2 becomes -sin theta, costheta 435 00:26:21.228 --> 00:26:24.054 So, by applying these transformed basis vectors 436 00:26:24.054 --> 00:26:28.687 multiplying each by x and y 437 00:26:28.687 --> 00:26:33.901 the original vector x, y in the initial space 438 00:26:33.901 --> 00:26:38.090 is transformed into a new vector in the rotated space of theta 439 00:26:38.090 --> 00:26:42.475 to determine its corresponding value 440 00:26:42.475 --> 00:26:44.374 This can be done using this formula 441 00:26:44.374 --> 00:26:47.927 So, a rotation by theta results in 442 00:26:47.927 --> 00:26:56.257 values of x cos theta - y sin theta, x sin theta + y cos theta 443 00:26:56.257 --> 00:27:02.287 These are the coordinates of the transformed vector 444 00:27:02.287 --> 00:27:05.989 For now, this formula simply shows 445 00:27:05.989 --> 00:27:10.713 how rotation works in spatial transformations 446 00:27:10.713 --> 00:27:14.036 As of right now, it is enough just to 447 00:27:14.036 --> 00:27:15.829 understand the concept 448 00:27:15.829 --> 00:27:20.624 I will continue to explain this formula further 449 00:27:20.624 --> 00:27:25.413 For now 450 00:27:25.413 --> 00:27:27.990 this is how a rotated vector is computed 451 00:27:27.990 --> 00:27:30.193 A vector rotated by theta 452 00:27:30.193 --> 00:27:31.949 can be calculated as follows 453 00:27:31.949 --> 00:27:37.178 I will summarize and explain it briefly this way 454 00:27:37.178 --> 00:27:42.465 Finally, let’s discuss inverse functions 455 00:27:42.465 --> 00:27:44.488 These are related to the previously mentioned 456 00:27:44.488 --> 00:27:49.356 sin, cos, and tan functions 457 00:27:49.356 --> 00:27:52.384 To invert these functions 458 00:27:52.384 --> 00:27:56.990 we swap the domain and codomain, making the codomain the domain 459 00:27:56.990 --> 00:28:01.163 and the original domain the codomain 460 00:28:01.163 --> 00:28:04.802 This reversed relationship is called the inverse function 461 00:28:04.802 --> 00:28:09.168 We might wonder if we can create such an inverse function 462 00:28:09.168 --> 00:28:12.178 However, the condition for creating an inverse function 463 00:28:12.178 --> 00:28:17.851 is that it must be a bijective function, as mentioned earlier 464 00:28:17.851 --> 00:28:20.733 But in the case of sin and cos 465 00:28:20.733 --> 00:28:22.356 they are not bijective functions 466 00:28:22.356 --> 00:28:23.350 Why? 467 00:28:23.350 --> 00:28:28.923 Currently, they are not one-to-one mappings 468 00:28:28.923 --> 00:28:31.594 For every x 469 00:28:31.594 --> 00:28:35.839 while y values range from -1 to 1 470 00:28:35.839 --> 00:28:39.317 if we limit y values to -1 and 1 471 00:28:39.317 --> 00:28:44.614 to make the codomain and range identical, they can be surjective 472 00:28:44.614 --> 00:28:48.666 However, for x = 1 473 00:28:48.666 --> 00:28:52.851 the corresponding x-values repeat infinitely 474 00:28:52.851 --> 00:28:54.599 This cannot be considered one-to-one 475 00:28:54.599 --> 00:28:57.822 Thus, these functions are not injective 476 00:28:57.822 --> 00:29:00.544 Since they are not bijective 477 00:29:00.544 --> 00:29:04.732 inverse functions cannot be created 478 00:29:04.732 --> 00:29:09.442 But to make them bijective 479 00:29:09.442 --> 00:29:12.059 and satisfy the property of injectivity 480 00:29:12.059 --> 00:29:15.213 if we deliberately restrict the domain values 481 00:29:15.213 --> 00:29:19.327 we can construct a bijective function within the restricted range 482 00:29:19.327 --> 00:29:23.317 This allows us to create inverse functions 483 00:29:23.317 --> 00:29:26.217 For the sin function 484 00:29:26.217 --> 00:29:29.307 the non-repeating, one-to-one interval 485 00:29:29.307 --> 00:29:33.257 is between -90 and 90 degrees 486 00:29:33.257 --> 00:29:36.543 For the cos function, let's take the positive direction 487 00:29:36.543 --> 00:29:39.099 It is between 0 and 180 degrees 488 00:29:39.099 --> 00:29:45.079 Within these ranges, they satisfy bijectivity 489 00:29:45.079 --> 00:29:51.158 For tan, since there’s no solution at -90 degrees 490 00:29:51.158 --> 00:29:52.950 it cannot be included 491 00:29:52.950 --> 00:29:56.253 So, this range is inclusive 492 00:29:56.253 --> 00:30:00.208 and this one excludes the boundary 493 00:30:00.208 --> 00:30:02.548 Within these excluded ranges 494 00:30:02.548 --> 00:30:05.465 we can construct bijective functions 495 00:30:05.465 --> 00:30:10.218 By restricting the range, we define inverse functions 496 00:30:10.218 --> 00:30:15.782 as arc sin, arc cos, and arc tan 497 00:30:15.782 --> 00:30:20.307 Thus, inverse functions may be denoted with a -1 superscript 498 00:30:20.307 --> 00:30:24.336 or names arc sin 499 00:30:24.336 --> 00:30:27.355 So, the inverse functions of these trigonometric functions 500 00:30:27.355 --> 00:30:30.525 can be constructed as follows 501 00:30:30.525 --> 00:30:36.841 Since these represent reversed mappings 502 00:30:36.841 --> 00:30:42.125 the domain intervals previously defined now become the range in the inverse function 503 00:30:42.125 --> 00:30:44.297 This applies to the codomain or range 504 00:30:44.297 --> 00:30:51.237 Thus, the intervals of the range are reversed 505 00:30:51.237 --> 00:30:56.079 For the range intervals, -90 to 90 degrees 506 00:30:56.079 --> 00:31:01.109 both the sin and tan functions have a range of -90 to 90 degrees 507 00:31:01.109 --> 00:31:06.025 Meanwhile, the cos function has a range of 0 to 180 degrees 508 00:31:06.025 --> 00:31:08.336 This corresponds to a range from 0 from pi radians 509 00:31:08.336 --> 00:31:13.237 This means that when we know a cos value 510 00:31:13.237 --> 00:31:16.079 or any sin, cos, or tan value 511 00:31:16.079 --> 00:31:20.435 we can determine the corresponding angle 512 00:31:20.435 --> 00:31:24.376 so this can be a useful tool for calculations 513 00:31:24.376 --> 00:31:26.686 The issue arises when finding these inverse functions 514 00:31:26.686 --> 00:31:29.475 We can obtain the angles 515 00:31:29.475 --> 00:31:33.582 but the values in the third and fourth quadrants 516 00:31:33.582 --> 00:31:37.059 cannot be determined 517 00:31:37.059 --> 00:31:38.515 These quadrant values remain unknown 518 00:31:38.515 --> 00:31:41.396 This is because the bijection condition must be satisfied 519 00:31:41.396 --> 00:31:43.295 Certain intervals are discarded 520 00:31:43.295 --> 00:31:47.396 and the values in these quadrants are excluded 521 00:31:47.396 --> 00:31:50.587 Therefore, using any trigonometric value 522 00:31:50.587 --> 00:31:55.940 we cannot infer the corresponding angles in these regions 523 00:31:55.940 --> 00:31:59.292 So, how can we infer these values? 524 00:31:59.292 --> 00:32:05.049 We can use an extension of the arctan function 525 00:32:05.049 --> 00:32:10.158 For arctan, it's defined as the ratio of the opposite side to the adjacent side 526 00:32:10.158 --> 00:32:12.484 Since tan is also the ratio of height to base 527 00:32:12.484 --> 00:32:18.673 when we reverse this and have the height-to-base ratio 528 00:32:18.673 --> 00:32:22.129 we can determine the angle from it 529 00:32:22.129 --> 00:32:28.564 Suppose the height-to-base ratio produces a positive value 530 00:32:28.564 --> 00:32:30.948 There are two cases of a positive value 531 00:32:30.948 --> 00:32:32.854 the base and height are 532 00:32:32.854 --> 00:32:36.732 both positive 533 00:32:36.732 --> 00:32:38.670 or both negative 534 00:32:38.670 --> 00:32:42.861 In either case, the result is positive 535 00:32:42.861 --> 00:32:46.644 So, when using tan values 536 00:32:46.644 --> 00:32:51.445 to find angles in reverse 537 00:32:51.445 --> 00:32:54.723 two possibilities always exist 538 00:32:54.723 --> 00:32:57.284 However, due to the inverse function’s constraints 539 00:32:57.284 --> 00:33:02.346 we can only retrieve values from this restricted region 540 00:33:02.346 --> 00:33:08.426 But if either x or y is negative, the situation changes 541 00:33:08.426 --> 00:33:12.614 If we consider the sign information 542 00:33:12.614 --> 00:33:17.079 and one of the values is negative 543 00:33:17.079 --> 00:33:19.663 then it must correspond 544 00:33:19.663 --> 00:33:23.373 to an angle in the third quadrant 545 00:33:23.373 --> 00:33:25.079 This conclusion can be drawn 546 00:33:25.079 --> 00:33:27.579 Thus, when creating an inverse function 547 00:33:27.579 --> 00:33:30.713 we don’t just provide the trigonometric value 548 00:33:30.713 --> 00:33:32.972 but also the sign of each component 549 00:33:32.972 --> 00:33:35.208 In other words, we provide both values 550 00:33:35.208 --> 00:33:41.128 With this, we can cover angles in all four quadrants 551 00:33:41.128 --> 00:33:43.396 We can determine the correct angles 552 00:33:43.396 --> 00:33:50.860 This is precisely the two-argument version of arctan 553 00:33:50.860 --> 00:33:53.703 known as the atan2 function 554 00:33:53.703 --> 00:33:59.645 In most trigonometric applications 555 00:33:59.645 --> 00:34:04.356 the atan2 function is frequently used 556 00:34:04.356 --> 00:34:06.879 Its first argument is y 557 00:34:06.879 --> 00:34:07.733 Remember this carefully 558 00:34:07.733 --> 00:34:10.883 The first argument is y, and the second is x 559 00:34:10.883 --> 00:34:13.852 It takes these two inputs 560 00:34:13.852 --> 00:34:17.195 To determine which quadrant the angle lies in 561 00:34:17.195 --> 00:34:20.922 it requires both arguments 562 00:34:20.922 --> 00:34:22.868 What this means is 563 00:34:22.868 --> 00:34:25.627 when calculating the ratio of height to base 564 00:34:25.627 --> 00:34:34.288 to find the value of a vector 565 00:34:34.288 --> 00:34:35.229 we use the ratio 566 00:34:35.229 --> 00:34:38.430 of x and y 567 00:34:38.430 --> 00:34:41.209 The base is x, and the height is y 568 00:34:41.209 --> 00:34:43.381 When x and y are known 569 00:34:43.381 --> 00:34:46.001 and these values are input into arctan 570 00:34:46.001 --> 00:34:50.516 it provides the angle of rotation 571 00:34:50.516 --> 00:34:56.110 relative to the origin 572 00:34:56.110 --> 00:34:59.199 In other words, the angle formed by a vector 573 00:34:59.199 --> 00:35:04.263 or a specific vector can be 574 00:35:04.263 --> 00:35:07.351 directly obtained using the highly useful function 575 00:35:07.351 --> 00:35:09.902 called atan2 576 00:35:09.902 --> 00:35:15.238 Thus, atan2 is extremely handy 577 00:35:15.238 --> 00:35:18.506 and versatile for angle calculations 578 00:35:18.506 --> 00:35:22.634 Understanding its principles will make it even more effective 579 00:35:22.634 --> 00:35:25.290 To understand why it requires two arguments 580 00:35:25.290 --> 00:35:27.685 Review this video later 581 00:35:27.685 --> 00:35:30.753 to deepen your understanding 582 00:35:30.753 --> 00:35:35.591 Finally, let’s briefly touch upon polar coordinates 583 00:35:35.591 --> 00:35:37.644 I’ll give a concise explanation 584 00:35:37.644 --> 00:35:44.488 In general, we express arbitrary points in the plane of a vector space 585 00:35:44.488 --> 00:35:50.476 as x,y combining their x- and y- values 586 00:35:50.476 --> 00:35:53.781 Instead, let’s consider it as being composed of circles 587 00:35:53.781 --> 00:35:55.872 The plane can essentially be viewed as a collection of circles 588 00:35:55.872 --> 00:35:58.852 It consists of infinitely many circles 589 00:35:58.852 --> 00:36:02.362 and the radius of each circle is denoted as R 590 00:36:02.362 --> 00:36:04.733 We use the radius to represent them 591 00:36:04.733 --> 00:36:10.874 We denote any point on the circle’s arc 592 00:36:10.874 --> 00:36:12.724 with an angle theta 593 00:36:12.724 --> 00:36:19.486 So, instead of expressing a vector in the plane as x, y 594 00:36:19.486 --> 00:36:23.021 we use the radius of the circle 595 00:36:23.021 --> 00:36:28.179 and the angle theta, which is known as the polar coordinate system 596 00:36:28.179 --> 00:36:31.862 The name is polar coordinate system 597 00:36:31.862 --> 00:36:36.130 The polar coordinate system is 598 00:36:36.130 --> 00:36:40.496 particularly useful for representing rotational effects 599 00:36:40.496 --> 00:36:45.083 Although Cartesian coordinates (x,y) are typically used 600 00:36:45.083 --> 00:36:47.387 to simplify rotational calculations 601 00:36:47.387 --> 00:36:50.128 we often convert temporarily 602 00:36:50.128 --> 00:36:53.773 to polar coordinates 603 00:36:53.773 --> 00:36:55.349 So, how do we convert from Cartesian coordinates 604 00:36:55.349 --> 00:36:58.565 to polar coordinates? 605 00:36:58.565 --> 00:37:02.533 As mentioned earlier, the radius 606 00:37:02.533 --> 00:37:06.602 is obtained by taking the square root of x^2 + y^2 607 00:37:06.602 --> 00:37:08.496 using the Pythagorean theorem 608 00:37:08.496 --> 00:37:12.424 This represents the norm or magnitude of the vector 609 00:37:12.424 --> 00:37:14.050 which is essentially its length 610 00:37:14.050 --> 00:37:17.298 Next, when we measure the angle from 0 degrees 611 00:37:17.298 --> 00:37:19.595 the angle can be determined 612 00:37:19.595 --> 00:37:25.417 using the atan function, as previously explained 613 00:37:25.417 --> 00:37:28.801 Input y and x into atan2 614 00:37:28.801 --> 00:37:31.882 and you can immediately get the angle 615 00:37:31.882 --> 00:37:34.056 Thus, Cartesian coordinates (x,y) 616 00:37:34.056 --> 00:37:37.466 can be converted into polar coordinates 617 00:37:37.466 --> 00:37:41.132 When polar coordinate values are given 618 00:37:41.132 --> 00:37:44.357 to convert them back to Cartesian coordinates 619 00:37:44.357 --> 00:37:50.311 for a radius R 620 00:37:50.311 --> 00:37:55.021 and an angle theta, as explained earlier 621 00:37:57.922 --> 00:37:59.144 the conversion is as follows 622 00:37:59.144 --> 00:38:05.258 The x-value is r cos theta, and the y-value is r sin theta 623 00:38:05.258 --> 00:38:11.846 So, if you calculate all the polar coordinate values 624 00:38:11.846 --> 00:38:14.724 and want to return to Cartesian coordinates 625 00:38:14.724 --> 00:38:18.532 then r cos theta corresponds to x 626 00:38:18.532 --> 00:38:20.545 and r sin theta corresponds to y 627 00:38:20.545 --> 00:38:22.533 This is how it works 628 00:38:22.533 --> 00:38:24.842 You can apply this transformation as needed 629 00:38:24.842 --> 00:38:28.832 Polar coordinates have many practical applications 630 00:38:28.832 --> 00:38:33.690 For instance, this transformation here 631 00:38:33.690 --> 00:38:38.575 it converts a shape originally composed of rectangles 632 00:38:38.575 --> 00:38:42.409 such that as the radius increases outward from the center 633 00:38:42.409 --> 00:38:44.139 it rotates more significantly 634 00:38:44.139 --> 00:38:47.619 If rotated uniformly, the rectangle simply spins around 635 00:38:47.619 --> 00:38:49.704 It maintains its rectangular shape while spinning 636 00:38:49.704 --> 00:38:53.070 But if you want the rotation to intensify as you move outward 637 00:38:53.070 --> 00:38:56.981 you can first convert to polar coordinates 638 00:38:56.981 --> 00:39:03.001 As r increases 639 00:39:03.001 --> 00:39:06.392 you can assign increasingly larger theta values 640 00:39:06.392 --> 00:39:10.674 creating a swirling effect at the edges 641 00:39:10.674 --> 00:39:15.080 This effect can be easily implemented using polar coordinates 642 00:39:16.460 --> 00:39:20.282 Linearity 643 00:39:20.516 --> 00:39:22.806 In mathematics, linearity 644 00:39:22.806 --> 00:39:25.139 is not about the shape of a line 645 00:39:25.139 --> 00:39:28.728 It refers to satisfying the following two equations 646 00:39:28.728 --> 00:39:32.545 which define linearity 647 00:39:32.545 --> 00:39:37.466 These are additivity and homogeneity 648 00:39:37.466 --> 00:39:40.548 Additivity means that for a given mapping 649 00:39:40.548 --> 00:39:46.040 if f(x+y) = f(x) + f(y) 650 00:39:46.040 --> 00:39:48.238 when inputs x and y are used 651 00:39:48.238 --> 00:39:54.229 and the results of f(x+y) equals f(x) + f(y) 652 00:39:54.229 --> 00:39:56.941 then the function satisfies additivity 653 00:39:56.941 --> 00:40:00.308 Secondly, for any scalar a 654 00:40:00.308 --> 00:40:03.615 if the result of a*f(x) 655 00:40:03.615 --> 00:40:07.434 equals f(ax) 656 00:40:07.434 --> 00:40:11.258 the function is said to have homogeneity 657 00:40:11.258 --> 00:40:15.526 If both additivity and homogeneity are satisfied 658 00:40:15.526 --> 00:40:19.644 the function is considered linear 659 00:40:19.644 --> 00:40:22.963 So, which mappings exhibit linearity? 660 00:40:22.963 --> 00:40:26.199 or we can ask which functions can be called linear 661 00:40:26.199 --> 00:40:31.040 Let’s explore which functions satisfy these properties 662 00:40:31.040 --> 00:40:37.927 For example, let’s consider the function y= x 663 00:40:37.927 --> 00:40:40.407 We’ll examine this function 664 00:40:40.407 --> 00:40:43.569 Let’s input two values 665 00:40:43.569 --> 00:40:45.169 5 and 10 666 00:40:45.169 --> 00:40:46.782 with 5 and 10 667 00:40:46.782 --> 00:40:51.219 x = 5 and y = 10 668 00:40:51.219 --> 00:40:55.714 Then, f(5+10) = f(15) 669 00:40:55.714 --> 00:41:01.694 If f(5+10) = f(5) + f(10), it satisfies additivity 670 00:41:01.694 --> 00:41:05.171 Next, for any scalar a 671 00:41:05.171 --> 00:41:08.694 say a = 5 672 00:41:08.694 --> 00:41:11.127 and x =10 673 00:41:11.127 --> 00:41:15.111 if 5*f(10) = f(50) 674 00:41:15.111 --> 00:41:17.842 then the function satisfies linearity 675 00:41:17.842 --> 00:41:22.268 We can say it satisfies linearity for these values 676 00:41:22.268 --> 00:41:26.694 Although this doesn’t guarantee satisfaction for all values 677 00:41:26.694 --> 00:41:30.080 we can rule out cases where it fails 678 00:41:30.080 --> 00:41:34.808 Let’s examine which functions satisfy these properties 679 00:41:34.808 --> 00:41:37.605 and which do not 680 00:41:37.605 --> 00:41:45.268 for y = x, or to make it easier 681 00:41:45.268 --> 00:41:50.595 we will use f(x) = x 682 00:41:50.595 --> 00:41:54.028 So, for two values 683 00:41:54.028 --> 00:41:58.466 5 and 10, since the output matches the input 684 00:41:58.466 --> 00:41:59.979 f(15) = 15 685 00:41:59.979 --> 00:42:04.159 and f(5) + f(10) = 15 686 00:42:04.159 --> 00:42:07.664 Since the results are the same, it satisfies additivity 687 00:42:07.664 --> 00:42:10.587 Next, 5*f(10) 688 00:42:10.587 --> 00:42:12.793 equals the result for f(50) 689 00:42:12.793 --> 00:42:16.417 it also satisfies homogeneity 690 00:42:16.417 --> 00:42:18.356 Thus, for the inputs 5 and 10 691 00:42:18.356 --> 00:42:22.357 we confirm that it satisfies linearity 692 00:42:22.357 --> 00:42:24.678 Now consider y = 2x 693 00:42:24.678 --> 00:42:27.446 Let's rewrite it as f(x) = 2x 694 00:42:27.446 --> 00:42:31.565 You can similarly rewrite the others as f(x) 695 00:42:31.565 --> 00:42:33.922 What about this function? 696 00:42:33.922 --> 00:42:36.259 For an input of 15 697 00:42:36.259 --> 00:42:38.813 the result is 2*15 698 00:42:38.813 --> 00:42:43.427 The sum of 2*5 and 2*10 also matches 699 00:42:43.427 --> 00:42:49.308 This equality follows from the distributive property 700 00:42:49.308 --> 00:42:53.302 Similarly, for the second property of homogeneity 701 00:42:53.302 --> 00:42:58.634 since the associative property holds for multiplication 702 00:42:58.634 --> 00:43:02.216 whether you multiply 5 later or rearrange the terms 703 00:43:02.216 --> 00:43:04.704 the result remains the same 704 00:43:04.704 --> 00:43:08.448 Thus, the function f(x) = 2x 705 00:43:08.448 --> 00:43:11.793 can be considered linear 706 00:43:11.793 --> 00:43:16.149 Now, let’s examine the function f(x) = 2x + 1 707 00:43:16.149 --> 00:43:21.713 If you input 15, the result is 708 00:43:21.713 --> 00:43:24.704 30 + 1 = 31 on the left side 709 00:43:24.704 --> 00:43:30.486 On the right side, it becomes 11 + 21 = 32 710 00:43:30.486 --> 00:43:34.189 Thus, the left and right sides are not equal 711 00:43:34.189 --> 00:43:36.842 Hence, it does not satisfy additivity 712 00:43:36.842 --> 00:43:40.634 This means the function cannot be considered linear 713 00:43:40.634 --> 00:43:42.714 We can definitively conclude this 714 00:43:42.714 --> 00:43:45.011 Because we have found a counterexample 715 00:43:45.011 --> 00:43:48.852 Let’s also look at y = x^2 716 00:43:48.852 --> 00:43:52.981 Here too, the left and right sides are not equal 717 00:43:52.981 --> 00:43:57.446 Similarly, it does not satisfy homogeneity 718 00:43:57.446 --> 00:44:00.664 We can see that it fails both properties 719 00:44:00.664 --> 00:44:08.456 From this, we can infer that a function of the form 720 00:44:08.456 --> 00:44:12.654 y=ax, where a coefficient is attached to a first-degree term 721 00:44:12.654 --> 00:44:17.060 is likely to satisfy linearity 722 00:44:17.060 --> 00:44:23.258 Let’s test the function y = ax 723 00:44:23.258 --> 00:44:27.474 or f(x) = ax 724 00:44:27.474 --> 00:44:31.446 to see if it satisfies linearity 725 00:44:31.446 --> 00:44:33.474 Here, for two arbitrary values 726 00:44:33.474 --> 00:44:38.476 say scalars b and c 727 00:44:38.476 --> 00:44:43.486 we need to check if f(b+c) = f(b) + f(c) 728 00:44:43.486 --> 00:44:46.558 Since the distributive property holds 729 00:44:46.558 --> 00:44:50.922 a*(b+c) becomes 730 00:44:50.922 --> 00:44:54.308 a*b + a*c on the left side 731 00:44:54.308 --> 00:44:57.059 and a*b + a*c on the right 732 00:44:57.059 --> 00:45:02.704 So by the field axioms, the distributive property ensures that 733 00:45:02.704 --> 00:45:04.275 we can confirm that 734 00:45:04.275 --> 00:45:06.664 the left and right sides are equal 735 00:45:06.664 --> 00:45:11.090 Thus, we see that it satisfies additivity 736 00:45:11.090 --> 00:45:14.862 Next, for an arbitrary scalar k 737 00:45:14.862 --> 00:45:19.367 let's check if k*f(b) = f(kb) 738 00:45:19.367 --> 00:45:23.658 In this case, substituting b and multiplying by k 739 00:45:23.658 --> 00:45:30.060 k*(ab) is the same as a*(kb) 740 00:45:30.060 --> 00:45:36.223 Since multiplication satisfies the commutative and associative properties 741 00:45:36.223 --> 00:45:38.404 we can confirm that 742 00:45:38.404 --> 00:45:41.922 this condition is also satisfied 743 00:45:41.922 --> 00:45:45.824 So, the function y = ax 744 00:45:45.824 --> 00:45:51.080 can be concluded to satisfy linearity 745 00:45:51.080 --> 00:45:54.733 However, when the function takes the form y=ax+b 746 00:45:54.733 --> 00:45:58.060 it does not satisfy linearity, as seen earlier 747 00:45:58.060 --> 00:46:01.758 Let’s verify if all functions of the form y=ax+b 748 00:46:01.758 --> 00:46:05.169 fail to satisfy linearity 749 00:46:05.169 --> 00:46:10.090 Here, the condition is that b is not 0 750 00:46:10.090 --> 00:46:14.772 Expanding the equation 751 00:46:14.772 --> 00:46:18.100 we are working with scalar fields 752 00:46:18.100 --> 00:46:20.077 So, using the properties of fields 753 00:46:20.077 --> 00:46:23.436 like commutative, associative, and distributive laws 754 00:46:23.436 --> 00:46:28.080 we find that the two sides differ 755 00:46:28.080 --> 00:46:30.837 On the left side, b appears only once 756 00:46:30.837 --> 00:46:33.116 while on the right side, b appears twice 757 00:46:33.116 --> 00:46:37.100 This discrepancy arises due to b 758 00:46:37.100 --> 00:46:39.966 Thus, as mentioned earlier, linearity 759 00:46:39.966 --> 00:46:42.575 was described as a property of lines 760 00:46:42.575 --> 00:46:46.172 While ax satisfies it 761 00:46:46.172 --> 00:46:51.110 ax+b does not, as we have seen 762 00:46:51.110 --> 00:46:53.927 If both are linear in form 763 00:46:53.927 --> 00:46:58.783 why doesn’t the second case satisfy linearity? 764 00:46:58.783 --> 00:47:03.334 Linearity, in its original sense 765 00:47:03.334 --> 00:47:06.268 doesn’t strictly refer to the shape of a line 766 00:47:06.268 --> 00:47:10.807 It signifies that two inputs are 767 00:47:10.807 --> 00:47:14.580 purely related in a first-degree proportional manner 768 00:47:14.580 --> 00:47:16.763 This describes a proportional mapping 769 00:47:16.763 --> 00:47:20.130 It’s not about judging by shape 770 00:47:20.130 --> 00:47:24.004 but rather the relationship between two inputs and their outputs 771 00:47:24.004 --> 00:47:28.750 being purely proportional in a linear relationship 772 00:47:28.750 --> 00:47:33.199 It’s better to understand it as such 773 00:47:33.199 --> 00:47:36.763 rather than focusing on the line 774 00:47:36.763 --> 00:47:39.542 In the case of y=ax + b 775 00:47:39.542 --> 00:47:43.881 the b term has no connection to the relationship between x and y 776 00:47:43.881 --> 00:47:46.238 It disrupts the input-output relationship 777 00:47:46.238 --> 00:47:50.465 as b is an unrelated, extraneous term 778 00:47:50.465 --> 00:47:53.872 This b, like an impurity 779 00:47:53.872 --> 00:47:56.561 breaks the pure linear proportionality 780 00:47:56.561 --> 00:48:01.080 so it cannot satisfy linearity 781 00:48:01.080 --> 00:48:05.451 Reinterpreting this 782 00:48:05.451 --> 00:48:07.999 when we ask what additivity means 783 00:48:07.999 --> 00:48:10.011 it can be explained as follows 784 00:48:10.011 --> 00:48:15.407 Think of water and oil, which don’t mix 785 00:48:15.407 --> 00:48:17.291 These two independent factors 786 00:48:17.291 --> 00:48:20.120 are like two separate elements 787 00:48:20.120 --> 00:48:25.936 When combined and processed through a machine 788 00:48:25.936 --> 00:48:28.912 we might not know the exact result 789 00:48:28.912 --> 00:48:31.585 but let’s assume it satisfies linearity 790 00:48:31.585 --> 00:48:33.508 When water and oil are mixed together 791 00:48:33.508 --> 00:48:36.872 and processed, there’s a resulting output 792 00:48:36.872 --> 00:48:40.259 If you instead process water alone, then oil alone 793 00:48:40.259 --> 00:48:43.724 you’d get separate outputs for each 794 00:48:43.724 --> 00:48:51.127 Combining these separate results 795 00:48:51.127 --> 00:48:55.664 should give the same result 796 00:48:55.664 --> 00:48:58.159 as processing water and oil together 797 00:48:58.159 --> 00:49:02.169 This means the result is purely based on water and oil 798 00:49:02.169 --> 00:49:04.258 with no extraneous factors 799 00:49:04.258 --> 00:49:08.516 There are no additional impurities 800 00:49:08.516 --> 00:49:11.951 Now, regarding homogeneity 801 00:49:11.951 --> 00:49:18.719 If the output is purely based on a combination of water and oil 802 00:49:18.719 --> 00:49:21.328 then the resulting change 803 00:49:21.328 --> 00:49:25.822 is not quadratic, cubic, or higher-order 804 00:49:25.822 --> 00:49:28.184 like x^2, x^3, x^4 805 00:49:28.184 --> 00:49:33.684 but purely proportional in a linear manner 806 00:49:33.684 --> 00:49:37.832 It reflects a purely first-order proportionality 807 00:49:37.832 --> 00:49:41.175 In other words, the output directly corresponds 808 00:49:41.175 --> 00:49:43.941 to the input 809 00:49:43.941 --> 00:49:46.828 Even if the input increases 810 00:49:46.828 --> 00:49:52.011 the output increases proportionally 811 00:49:52.011 --> 00:49:56.130 So, with these two properties 812 00:49:56.130 --> 00:49:59.461 you can predict the outcomes 813 00:49:59.461 --> 00:50:03.585 of each independent factor 814 00:50:03.585 --> 00:50:07.335 By separating and analyzing them individually 815 00:50:07.335 --> 00:50:13.373 then combining the results later 816 00:50:13.373 --> 00:50:15.605 you can predict the final outcome 817 00:50:15.605 --> 00:50:18.912 This is called the principle of superposition 818 00:50:18.912 --> 00:50:24.076 When two independent patterns exist 819 00:50:24.076 --> 00:50:27.120 and they satisfy linearity 820 00:50:27.120 --> 00:50:30.387 even if combined in a complex way 821 00:50:30.387 --> 00:50:32.628 you can handle them simply 822 00:50:32.628 --> 00:50:35.684 by analyzing each separately 823 00:50:35.684 --> 00:50:39.631 as the combined result will be the same 824 00:50:39.631 --> 00:50:42.916 This allows for modular, step-by-step processing 825 00:50:42.916 --> 00:50:45.981 You can interpret it this way 826 00:50:45.981 --> 00:50:48.209 Linearity is incredibly useful 827 00:50:48.209 --> 00:50:52.862 when analyzing phenomena 828 00:50:52.862 --> 00:50:56.947 In graphics, linearity is extensively used 829 00:50:56.947 --> 00:50:59.040 in transformations 830 00:50:59.040 --> 00:51:02.892 So let’s delve into transformations 831 00:51:02.892 --> 00:51:07.967 The key term you need to know for this 832 00:51:07.967 --> 00:51:10.644 is linear mapping 833 00:51:10.644 --> 00:51:13.565 It is linear mapping in English 834 00:51:13.565 --> 00:51:17.508 Previously, linearity 835 00:51:17.508 --> 00:51:21.278 referred to properties of a function f(x) 836 00:51:21.278 --> 00:51:26.189 So, what’s the difference between a mapping and a function? 837 00:51:26.189 --> 00:51:28.332 A function, as mentioned earlier 838 00:51:28.332 --> 00:51:31.849 is a relationship between sets 839 00:51:31.849 --> 00:51:33.634 A mapping is similar 840 00:51:33.634 --> 00:51:36.518 It also describes relationships between sets 841 00:51:36.518 --> 00:51:38.172 but within a mathematical framework 842 00:51:38.172 --> 00:51:42.763 like field axioms or vector space axioms 843 00:51:42.763 --> 00:51:44.958 Mappings preserve these structures 844 00:51:44.958 --> 00:51:49.605 while relating sets 845 00:51:49.605 --> 00:51:52.686 Based on this mapping 846 00:51:52.686 --> 00:51:56.565 when we input a vector 847 00:51:56.565 --> 00:52:00.384 the output 848 00:52:00.384 --> 00:52:03.027 retains the original system’s properties 849 00:52:03.027 --> 00:52:06.486 This is what we call a mapping 850 00:52:06.486 --> 00:52:09.514 In both vector spaces and real number spaces 851 00:52:09.514 --> 00:52:11.824 the system of field axioms 852 00:52:11.824 --> 00:52:14.436 remain intact 853 00:52:14.436 --> 00:52:20.704 From this perspective, it can be considered a mapping 854 00:52:20.704 --> 00:52:25.813 Here, it can be seen as a binary operation 855 00:52:25.813 --> 00:52:28.636 Both x and y are scalars 856 00:52:28.636 --> 00:52:30.643 These two values, through some function 857 00:52:30.643 --> 00:52:33.793 combine additively to produce a result 858 00:52:33.793 --> 00:52:35.829 If scalar values are input 859 00:52:35.829 --> 00:52:39.291 the structure of the field is preserved 860 00:52:39.291 --> 00:52:43.001 so it can be called a mapping 861 00:52:43.001 --> 00:52:47.808 If we have multiple inputs 862 00:52:47.808 --> 00:52:51.531 we can adjust the number of inputs and outputs 863 00:52:51.531 --> 00:52:55.872 as needed 864 00:52:55.872 --> 00:52:58.111 Mapping to a vector space means 865 00:52:58.111 --> 00:53:00.883 that the eight axioms of vector spaces we previously discussed 866 00:53:00.883 --> 00:53:03.902 are preserved 867 00:53:03.902 --> 00:53:07.486 This can also be called a mapping 868 00:53:07.486 --> 00:53:12.707 Now, let’s visually explore 869 00:53:12.707 --> 00:53:18.120 an example of a mapping in a vector space 870 00:53:18.120 --> 00:53:20.951 In a two-dimensional vector space 871 00:53:20.951 --> 00:53:25.862 let’s examine a mapping from R2 to R2 872 00:53:25.862 --> 00:53:28.951 Using an input 873 00:53:28.951 --> 00:53:31.327 that is a vector element of the vector space 874 00:53:31.327 --> 00:53:34.827 say x, y 875 00:53:34.827 --> 00:53:40.179 let’s consider how these transform into a new vector 876 00:53:40.179 --> 00:53:42.099 You can construct such a function 877 00:53:42.099 --> 00:53:45.714 We can imagine such a vector function 878 00:53:45.714 --> 00:53:49.948 If it maps R2 to R3 879 00:53:49.948 --> 00:53:53.944 using various forms of x and y 880 00:53:53.944 --> 00:53:59.258 you could generate a three-dimensional vector mapping 881 00:53:59.258 --> 00:54:02.168 Among these mappings 882 00:54:02.168 --> 00:54:06.644 there are mappings that possess linearity, as described earlier 883 00:54:06.644 --> 00:54:10.884 These will also follow a specific pattern 884 00:54:10.884 --> 00:54:14.763 What is a mapping that possesses linearity? 885 00:54:14.763 --> 00:54:17.804 For example, in a vector space 886 00:54:17.804 --> 00:54:21.316 if the result of adding two vectors as input 887 00:54:21.316 --> 00:54:26.545 equals the sum of the individual outputs 888 00:54:26.545 --> 00:54:29.355 it satisfies additivity 889 00:54:29.355 --> 00:54:36.080 This means if you multiply the result of inputting a vector 890 00:54:36.080 --> 00:54:39.204 with a scalar 891 00:54:39.204 --> 00:54:41.654 it equals the result of multiplying the scalar 892 00:54:41.654 --> 00:54:44.223 with the vector before input 893 00:54:44.223 --> 00:54:47.238 This is called homogeneity 894 00:54:47.238 --> 00:54:50.495 To satisfy this 895 00:54:50.495 --> 00:54:53.155 as mentioned earlier, linearity 896 00:54:53.155 --> 00:54:57.788 means that the input-output relationship 897 00:54:57.788 --> 00:55:01.080 is purely linear 898 00:55:01.080 --> 00:55:05.605 In other words, aside from the input values 899 00:55:05.605 --> 00:55:07.325 no extraneous factors should interfere 900 00:55:07.325 --> 00:55:10.938 The relationship must remain purely linear 901 00:55:10.938 --> 00:55:13.397 That’s the definition 902 00:55:13.397 --> 00:55:15.905 Let’s consider this function first 903 00:55:15.905 --> 00:55:21.698 When we input a vector (x,y) 904 00:55:21.698 --> 00:55:24.842 it produces a two-dimensional vector as a result 905 00:55:24.842 --> 00:55:28.824 Here, x is represented as x + 1 906 00:55:28.824 --> 00:55:33.397 and y remains unchanged in this vector function 907 00:55:33.397 --> 00:55:36.638 However, x + 1 includes 908 00:55:36.638 --> 00:55:40.160 a term that is unrelated to the x and y inputs 909 00:55:40.160 --> 00:55:41.991 which is 1, an extraneous factor 910 00:55:41.991 --> 00:55:44.704 Thus, it will not satisfy linearity 911 00:55:44.704 --> 00:55:46.924 Let’s test this by substituting values 912 00:55:46.924 --> 00:55:51.083 Using two vectors (3,6) and (4,8) for x,y 913 00:55:51.083 --> 00:55:53.667 if we add them 914 00:55:53.667 --> 00:55:58.308 We’ll calculate the result of adding and then inputting 915 00:55:58.308 --> 00:56:04.195 Next, we’ll input these two vectors separately 916 00:56:04.195 --> 00:56:07.650 The results of individually inputting them 917 00:56:07.650 --> 00:56:10.367 must match for additivity to hold 918 00:56:10.367 --> 00:56:13.876 adding them gives (7, 14) 919 00:56:13.876 --> 00:56:16.466 it results in (7, 14) 920 00:56:16.466 --> 00:56:21.080 while individual inputs give (4,6) and (5,8) 921 00:56:21.080 --> 00:56:27.530 Inputting (7,14) gives (8,14) 922 00:56:27.530 --> 00:56:32.229 but adding the individual results gives (9,14), which are not the same 923 00:56:32.229 --> 00:56:34.253 If these two matched 924 00:56:34.253 --> 00:56:36.524 additivity would have been satisfied 925 00:56:36.524 --> 00:56:39.963 Since the results do not match 926 00:56:39.963 --> 00:56:41.605 it does not satisfy additivity 927 00:56:41.605 --> 00:56:44.058 If additivity fails, there’s no need to check further 928 00:56:44.058 --> 00:56:47.555 it does not satisfy linearity 929 00:56:47.555 --> 00:56:54.538 Now, consider the first output as 3x + 4y 930 00:56:54.538 --> 00:56:58.268 and the second output as 2x+3y 931 00:56:58.268 --> 00:57:02.823 If the second element corresponds to 2x+3y 932 00:57:02.823 --> 00:57:06.973 there are no extraneous terms here 933 00:57:06.973 --> 00:57:09.567 There are no +1 or +2 terms 934 00:57:09.567 --> 00:57:12.951 nor are there squared or higher-order terms 935 00:57:12.951 --> 00:57:15.134 It’s composed of a purely linear relationship 936 00:57:15.134 --> 00:57:18.168 So, what about this case? 937 00:57:18.168 --> 00:57:21.922 We might think it satisfies additivity 938 00:57:21.922 --> 00:57:25.446 When we substitute two vectors 939 00:57:25.446 --> 00:57:28.667 the left and right sides produce the same result 940 00:57:28.667 --> 00:57:32.347 indicating it satisfies linearity 941 00:57:32.347 --> 00:57:43.455 Thus, for purely linear relationships 942 00:57:43.455 --> 00:57:46.250 with scalars added directly 943 00:57:46.250 --> 00:57:48.566 functions without extraneous terms 944 00:57:48.566 --> 00:57:52.080 of the form ax+by and cx+dy 945 00:57:52.080 --> 00:57:56.352 we can verify whether they always satisfy linearity 946 00:57:56.352 --> 00:57:59.930 Let’s take two vectors 947 00:57:59.930 --> 00:58:05.090 denote them as (x1, y1), and (x2, y2) 948 00:58:05.090 --> 00:58:08.282 and proceed with this equation 949 00:58:08.282 --> 00:58:10.707 When actually performing this calculation 950 00:58:10.707 --> 00:58:13.959 let’s check if linearity is satisfied 951 00:58:13.959 --> 00:58:16.459 According to the axioms of vector spaces 952 00:58:16.459 --> 00:58:20.570 the result of adding and then inputting two vectors 953 00:58:20.570 --> 00:58:23.741 is as follows 954 00:58:23.741 --> 00:58:26.747 Separately inputting and then adding gives 955 00:58:26.747 --> 00:58:29.207 the following form 956 00:58:29.207 --> 00:58:34.087 Since the distributive property is satisfied in vector spaces 957 00:58:34.087 --> 00:58:37.428 the scalar-vector distributive property holds 958 00:58:37.428 --> 00:58:42.019 Thus, these terms are equal 959 00:58:42.019 --> 00:58:45.405 Hence, it satisfies additivity 960 00:58:45.405 --> 00:58:48.863 Next, consider an arbitrary scalar k 961 00:58:48.863 --> 00:58:51.552 Let’s check if it satisfies homogeneity 962 00:58:51.552 --> 00:58:55.504 Similarly, using the axioms of vector spaces 963 00:58:55.504 --> 00:59:00.018 we can confirm that this property is satisfied 964 00:59:00.018 --> 00:59:05.038 Therefore, in any two-dimensional vector space 965 00:59:05.038 --> 00:59:07.557 a mapping from a 2D vector space to another 2D vector space 966 00:59:07.557 --> 00:59:14.678 like f(x,y) = (ax+by, cx+dy) 967 00:59:14.678 --> 00:59:20.444 can be concluded to satisfy linearity 968 00:59:20.444 --> 00:59:24.462 Thus, as mentioned earlier, linear mappings 969 00:59:24.462 --> 00:59:28.204 are the relationships between two structures or systems 970 00:59:28.204 --> 00:59:31.244 that preserve linearity 971 00:59:31.244 --> 00:59:35.861 If limited to vector spaces 972 00:59:35.861 --> 00:59:40.266 it gives the impression of space transformation 973 00:59:40.266 --> 00:59:43.824 More specifically, we use the term linear transformation 974 00:59:43.824 --> 00:59:45.552 to describe it 975 00:59:45.552 --> 00:59:50.579 If linear mapping is the term used 976 00:59:50.579 --> 00:59:54.937 then the term linear transformation 977 00:59:54.937 --> 00:59:57.286 is used for practical applications in vector spaces 978 00:59:57.286 --> 01:00:01.304 In fact, linear functions, mappings, and transformations 979 01:00:01.304 --> 01:00:05.184 can be considered to mean the same thing 980 01:00:05.184 --> 01:00:09.163 Which term should we use among these three? 981 01:00:09.163 --> 01:00:14.063 For vector spaces 982 01:00:14.063 --> 01:00:16.364 using the term linear transformation 983 01:00:16.364 --> 01:00:19.355 conveys the meaning more clearly 984 01:00:19.355 --> 01:00:22.408 Therefore, in vector spaces 985 01:00:22.408 --> 01:00:24.435 linear mappings are also referred to as linear transformations 986 01:00:24.435 --> 01:00:26.849 From now on, we will refer to such relationships 987 01:00:26.849 --> 01:00:29.191 as linear transformations 988 01:00:29.191 --> 01:00:34.333 Linear transformations in the 2D real vector space 989 01:00:34.333 --> 01:00:40.484 will have the following form 990 01:00:40.484 --> 01:00:43.724 Based on this, in the next session 991 01:00:43.724 --> 01:00:46.354 we will explore how to perform linear transformations 992 01:00:46.354 --> 01:00:49.414 and learn their methods 993 01:00:49.414 --> 01:00:51.457 This concludes today’s lecture 994 01:00:51.457 --> 01:00:53.192 Thank you for attending the lesson 995 01:00:53.192 --> 01:00:54.108 Thank you 996 01:00:54.724 --> 01:00:56.553 Trigonometric Functions Trig Ratios: The ratio of 2 sides among 3 sides of a right triangle Trig Functions: generalized relationship of trig ratios 997 01:00:56.553 --> 01:00:58.380 Angle Measurement Degree System: Represents angles using 360, a number with many divisors Radian System: Defines basic angle where the arc length is 1 998 01:00:58.380 --> 01:01:00.351 Trigonometry in Computer Graphics: Operates based on radians Vector Rotation: Requires understanding the mechanism of rotating an object by theta 999 01:01:00.351 --> 01:01:01.628 Rotation Transformation: Transforms two basis vectors while preserving their magnitude of 1, orthogonality, and current orientation 1000 01:01:01.638 --> 01:01:04.539 Trigonometric Functions Inverse Trigonometric Functions Since sin and cos are not bijective, their domains are restricted to create inverse functions 1001 01:01:04.539 --> 01:01:07.044 Can obtain the value of the angle formed by an arbitrary vector by using the arctan function Polar Coordinates Used instead of Cartesian coordinates for easier rotation handeling 1002 01:01:07.044 --> 01:01:08.687 Polar Coordinates: Represent positions using a circle's radius and the angle formed by the arc 1003 01:01:08.687 --> 01:01:10.400 Linearity Achieved if a function satisfies both additivity and homogeneity of degree 1 Additivity:f(x+y)=f(x)+f(y) Homogeneity of degree 1:af(x)=f(ax) 1004 01:01:10.400 --> 01:01:11.994 Additivity: result of combining 2 independent elements is same as processing separately and adding Homogeneity: pure linear proportional relationship 1005 01:01:11.994 --> 01:01:13.440 Linear mapping Mapping: a relation that corresponds to each other while preserving the mathematical system A linearity mapping preserves the system when a vector is applied to it 1006 01:01:13.440 --> 01:01:14.598 Linear Mapping: A correspondence between two structures with linearity Linear Transformation: A correspondence within a practical vector space