[图形学]3D数学基础_图形与游戏开发(习题答案)

Answers Chapter 2

1)Give the coordinates of the following points:

a (-2.5, 3)

b (1, 2)

c (2.5, 2)

d (-1, 1)

e (0, 0)

f (2, -0.5)

g (-0.5, -1.5)

h (0, -2)

j (-3, -2)

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2)List the 48 different possible ways that the 3D axes may be assigned to the

directions “north,” “east” and “up.” Identify which of these combinations are left-handed, and which are right-handed.

North East Up Hand North East Up Hand

+x +y +z Left +x +z +y Right

+x +y –z Right +x +z –y Left

+x –y +z Right +x –z +y Left

+x –y –z Left +x –z –y Right

–x +y +z Right –x +z +y Left

–x +y –z Left –x +z –y Right

–x –y +z Left –x –z +y Right

–x –y –z Right –x –z –y Left

+y +x +z Right +y +z +x Left

+y +x –z Left +y +z –x Right

+y –x +z Left +y –z +x Right

+y –x –z Right +y –z –x Left

–y +x +z Left –y +z +x Right

–y +x –z Right –y +z –x Left

–y –x +z Right –y –z +x Left

–y –x –z Left –y –z –x Right

+z +x +y Left +z +y +x Right

+z +x –y Right +z +y –x Left

+z –x +y Right +z –y +x Left

+z –x –y Left +z –y –x Right

–z +x +y Right –z +y +x Left

–z +x –y Left –z +y –x Right

–z –x +y Left –z –y +x Right

–z –x –y Right –z –y –x Left

3)In a popular modeling program 3D Studio Max, the default orientation of the axes is

for +x to point right, +y to point forward, and +z to point up. Is this a left- or right-handed coordinate space?

Right-handed.

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Chapter 3

1)Draw a nested space hierarchy tree for the sheep described in Section 3.3,

assuming that its head, ears, upper legs, lower legs, and body move indemendently.

2)Suppose our object axes are transformed to world axes by rotating them

counterclockwise around the y-axis by 42o and then translating six units along the

z-axis and 12 units along the x-axis. Describe this transformation from the

perspective of the object.

Imagine a point on the object, in object space. As the axes are rotating

counterclockwise, the point is actually rotating counterclockwise relative to the axes.

Then, as the axes translate by [12, 0, 6], the point translates [-12, 0, -6] relative to

the axes.

3)Which coordinate space is the most appropriate in which to ask the following

questions?

a)Is my computer in front of or behind me? Object space. If we know the

position of the computer within our object space, this question is a trivial

matter of checking for a positive z value. (Assuming the conventions from

Section 2.3.4)

b)Is the book east or west of me? Inertial space is the easiest space to make this

test. Again, assuming the conventions from Section 2.3.4, the book is east of

us if the x-coordinate of the book’s position in our inertial space is positive,

and west if this value is negative. Alternatively, we could answer the

question in world space, by comparing the x-coordinate of the book in world

space, with our own world space x-coordinate.

c)How do I get from one room to the other? Pathfinding-type querries are

usually made in world space.

d)Can I see my computer? The “camera space” for our viewpoint is the most

natural coordinate space to use for this question.

Chapter 4

1)Let:

a)Identify a, b, and c, as row or column vectors, and give the dimension of

each vector.

a is a 2D row vector.

b is a 3D column vector.

c is a 4D column vector.

b)Compute b y+c w+a x+b z.

2)Identity the quantities in each of the following sentences as scalar or vector. For

vector quantities, give the magnitude and direction. (Note: some directions may be

implicit.)

a)How much do you weigh? Weight is a calar quantity.

b)Do you have any idea how fast you were going? Speed is a scalar quantity.

c)It’s two blocks north of here. “Two blocks north” is a vector quantity, since

it specified a magnitude (“two blocks”) and a direction (“north”).

d)We’re cruising from Los Angeles to New York at 600mph, at an altitu de of

33,000ft. Speed (600mph) is a scalar quantity. However, since we know we

are traveling from Los Angeles to New York, we could assume an eastward

direction, which would provide a direction, making it a velocity, which is a

vector quantity. Altitude (33,000ft) is a scalar quantity.

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3)Give the values of the following vectors:

a [0, 2]

b [0, -2]

c [0.5, 2]

d [0.5, 2]

e [0.5, -3]

f [-2, 0]

g [-2, 1]

h [2.5, 2]

j [6, 1]

4)Identify the following statements as true or false. If the statement is false, explain

why.

a)The size of a vector in a diagram doesn’t matter; we just need to draw it

in the right place. False. Size matters; so does direction. A vector does not

express a “position,” and so we can draw in on a diagram anywh ere that is

convenient. See Section 4.2.2.

b)The displacement expressed by a vector can be visualized as a sequence

of axially aligned displacements.True. See Figure 4.5 on page 40.

c)These axially aligned displacements from the previous question must occur

in order. False. They can occur in any order, due to commutative nature of

vector addition. See page 40.

d)The vector [x, y] gives the displacement from the point (x, y) to the origin.

False. It gives the opposite displacement – from the origin to the point.