Question

1. Matter of Shading One way to accurately render three-dimensional objects on a computer screen involves using the dot and cross products. In order to determine how to shade a piece of a surface we need to determine the angle at which rays from the light source hit the surface. To determine this angle, we compute the dot product of the light vector with the vector perpendicular to the surface at the particular point, called the normal vector normal vector light ray normal vecor light ray This picce appcars bright This picce appcars dimmer of 180° with the If the light ray hits the surface surface will appear bright. On the other hand, if the light comes in on an angle, this piece of the surface will not appear as bright (a) Suppose the light source is placed directly above the xy-plane, so that the light rays come in parallel to the vector (0,0, 1. At what angle (to the normal) do the light rays hit a triangle bounded by the points (3, 2, 4), (2,5, 3), and (1,2,6)? of the on, that is, has then this Note that your answer will be an obtuse angle (greater than 90°) because the vertex of the angle is formed by the tails of the two vectors (b) Repeat Problem 1, this time using the region bounded by (3,5, 2), (3, 3, 1), and (1,3,1) (c) Suppose we are standing above the light source looking down on the xy-plane. Which of these two regions will appear brighter to us?

Answer 1

Gfven verti ces The angle o (3, Let e are P(3,2, u) & (2, 5, R ,2, 2- 1, 3, -1> 2-3, 5-2 3-u -- PR 1-3 6- u -2, o, 2> 2-2 A - 1 2 - 2 O (- 2-2 iaht ay Cores im Given Wrat -> Paoallel he

A Sltalat Argles betuoren Yay and o mal vectoy i\ 4, u, b 0,0,-1>, co Cosea.b O 129 762 129-76 introngle AC3, B, 2),53) verticcs o he and ,) A3 23-3, 3-5, -2 -2, AC -2 Ac i-3, 3-5, ^-2 -2 , -1>

Nosmal euoy k A A - } -9 -2 Co -2) 2-2 A -vecto Parallel -1> hua agle bct ween he t plare en by Lhe CoG -1> 26.57

Since O- 29- 76 26. 57 e ion yhe foints C3, eaion ien in fark (1) i afpcar to