Question 1
(a) Show that the following inequality is true for the two vectors u and v, u = (1, 1, 1) and v = (-1, -1, 3) and discuss the geometrical meaning of the result.
||u + v|| ≤ ||u|| + ||v||
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(b) Consider the vector u = (1/ √3, 1/ √3, 1/ √3) and vector v = (1/ √6, 1/ √6, -2/ √6). Discuss the two dot products u?u and u?v and the geometrical meaning of the results.
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Question 2
(a) A square object is defined by the homogeneous coordinates (-1, 1, 1, 1), (-1, 0, 1, 1), (0, 0, 1, 1) and (0, 1, 1, 1). Suppose the object is scaled 3 units in the x direction, 2 units in the y direction, and 1 unit in the z direction. Solve the transformation operation to determine the new position of the object.
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(b) Instead of performing the scaling operations, suppose the square object in (a) is rotated 90-degree counter clockwise about the x-axis. Solve the transformation operation to determine the new position of the object.
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(c) The rotated square in (b) is further rotated 90-degree counter clockwise about the z-axis. Solve the transformation operation to determine the final position of the object.
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(d) Solve the transformation to determine the matrix representing the two operations in (b) and (c).
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Question 3
(a) Suppose a surface is defined by the equation x/ √3 + y/ √3 + z/ √3 - √3 = 0. Is the point (1, 1, 1) lying on the surface? Describe the normal vector at a point (x, y, z) on the surface. Describe and calculate the unit normal vector at the point (1, 1, 1) if it lies on the surface. Would the unit normal vector be the same at other points on the surface?
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(b) Suppose another plane has unit normal vector (1/ √6, 1/ √6, 2/ √6) and is at a signed distance of 13/√6 from the origin. Suppose further that a ray passes through the point (-1, -1, -1), followed by the point (-2.732, -4.464, -6.196). Calculate the intersection point of the ray and the plane.
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Question 4
(a) Suppose you are doing an animation for a science fiction thriller. Your hero has been trapped in pitch-black pool filled with oil-like liquid in Planet M. He is trying hard to find his way but time is running out - it will only be minutes before his brother is put to eternal rest in the castle! He will need to let the allies know where he is. Unfortunately, in the total darkness he could not move fast. The liquid surrounding him is also dragging him further and further downwards. In his despair he suddenly remembers the sabre the sorcerer has given him. He frantically gropes for the sword and swings it wildly. And then there is light ...
Suppose the light ray is in the direction of (1/ √3, 1/ √3, 1/ √3). Calculate the refraction vector when the ray leaves the oil-like liquid. You may assume that the laws of physics on Planet M are the same as those in Planet Earth. You may also assume that the ratio of refractive index, between the liquid and the air outside, is 0.75, and that the normal vector of the surface of the pool is in the direction (0, 0, -1).
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(b) The light in the sabre is short-lived. It has gone off in just a few seconds. But it has also served its purpose. The sorcerer has detected his presence. After a short spell of darkness, your hero could see from within the pool another ray of light above the surface sent by the sorcerer ...
Suppose the ray of light in the direction (-1/ √6, -1/ √6, -2/ √6) hits the mirror-like flat surface of the castle besides the pool. Suppose further that the normal vector of the surface is in the direction (1/ √7, √2/ √7, 2/ √7). Compute the reflection vector.
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(c) The allies positioned around the pool finally see the castle and move towards there. They would like to move as fast as possible. Unfortunately, only some of them see the reflection of the sorcerer's light clearly and could move faster. Others positioned at more awkward locations could only see a dim spot and took a much longer time ...
Explain the Phong Reflection Model. Demonstrate, using this model, that the reflection is the brightest in when the allies are positioned in the same direction as the reflected ray. Demonstrate further the effect on the reflection intensity, for allies located at 60 degree away from the reflected ray, if the shininess of the castle surface has been changed from 2 to 4. You may assume that the specular reflection is 0.8, and the light source is a directional light with intensity 100.