Homework #7: 3D Rendering Geometry Solution

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Questions 1, 3, and 4 will walk you through the numerical computations for Lab 7.

 

Question 2 reviews the Gram-Schmidt orthogonalization we talked about in class.

 

 

 

 

 

  1. A vertex of a polygonal model is located at (10; 15; 0) in the model’s object space. The model is rotated by 45 degrees around the y-axis and placed so that it is centered at (its origin maps to) position (30; 0; 40) in the world.

 

  • Write out the model’s object-to-world transformation as a sequence of matrix operations. (You do not have to multiply out the matrices. You may also leave your answer in terms of trig functions.)

 

  • Where is that vertex now in world coordinates?

 

 

 

  1. A camera is located at position (25; 20; 5) in the 3D world and is looking at the point (25; 40; 25) so that the direction [0; 1; 0] points (roughly!) up.

 

  • Use the process we covered in class (a 3D variant of Gram-Schmidt orthogonalization using cross products) to calculate the camera’s x, y, and z axis directions.

 

  • Write this camera’s world-to-camera transformation as the composition of a rotation matrix and translation matrix. (You again do not have to multiply out this matrix.)
  • What are the camera-space coordinates of the point pw = (5; 6; 7)?

 

 

 

  1. A camera is located at position (20; 5; 40) and oriented so that it is pointing parallel to the x-z plane at an angle of 30 degrees off the z axis. (This is the basic setup for Labs 5–7.)

 

  • Write this camera’s world-to-camera transformation using the composition of a 3D rotation matrix (around the y axis) and a translation matrix. (You again do not have to multiply out this matrix. You may also leave your answer in terms of trig functions.)
  • What are the camera-space coordinates of the point pw = (5; 6; 7)?

 

 

  1. A virtual camera has the following parameters:

 

vertical field of view of 60 degrees

 

aspect ratio of 16:9 (horizontal to vertical) near plane n = 10

 

far plane f = 1000

 

  • What is the clip matrix for this camera?

 

  • What are the clip-space coordinates of the camera-space point pc = (5; 5; 50)?

 

  • Is this point pc = (5; 5; 50) within the view frustum of this camera? How can you tell directly from the clip-space coordinates without doing a division operation?

 

  • What are the canonical coordinates of this point pc = (5; 5; 50)?

 

  • If rendered to a high-definition display (1920 1080), what are the screen coordinates of this point?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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