计算机图形学（第三版）DonaldHearn蔡士杰译课后习题答案8

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Chapter 8

Three-Dimensional Object

Representations

This chapter examines the methods that have been devised to describe and visualize various

kinds of objects and data sets. In addition, programming projects explore the application of

OpenGL functions for diﬀerent representations, as well as the implementation details involved

in the algorithms for generating displays of splines, fractals, and other objects.

Exercises

8-1. Input for this algorithm is the sphere radius, assuming that the sphere is centered at the

coordinate origin. A polygon-mesh representation can be constructed using longitude and

latitude lines to obtain vertex positions. Thus, two additional input parameters could

be provided to specify the number of latitude and longitude lines that are to be used in

subdividing the sphere surface.

The parametric equations for either a longitude line or a latitude line can be obtained

from Eqs. 8-2. Along each longitude line, angle θ has a constant value. Similarly, angle

φ is constant along any latitude line. For example, the coordinates along the longitude

line in the xz plane (θ = 0) are calculated with the equations

x = r cos φ, y =0,z= r sin φ

and the equations for computing coordinate positions along the latitude line in the xy

plane (φ = 0) are

x = r cos θ, y = r sin θ, z =0

If we denote the required number of latitude lines as n

lat

and the required number of

longitude lines as n

long

, the algorithm can obtain vertex coordinates by calculating n

long

positions along n

lat

latitude lines (or vice versa). Thus, angle θ is assigned n

long

values

over the range from −π to +π, and angle φ is assigned n

lat

values over the range from

−π/2to+π/2.

48 CHAPTER 8. THREE-DIMENSIONAL OBJECT REPRESENTATIONS

The algorithm should take sphere symmetries into account to reduce the number of

trigonometric calculations. Coordinate positions need to be computed from the paramet-

ric sphere equations only for the latitude lines in the northern (or southern) hemisphere.

Positions in the other hemisphere are obtained using a reﬂection about the xy plane.

Also, the algorithm need only calculate positions within one octant of each circular lat-

itude (or longitude) line. The remaining positions in the other octants for each circle

are obtained by symmetry (Fig. 3-18). Furthermore, the latitude “lines” at the top and

bottom of the sphere (the poles) are points.

At the top of the sphere, polygon edges are deﬁned from the pole position radially down

to the coordinate positions on the ﬁrst latitude line below the pole, and a polygon edge

is deﬁned between each pair of adjacent positions along that latitude line. A similar set

of polygons is constructed at the bottom of the sphere. Thus, the top and bottom of the

spherical surface are each represented with a triangular mesh.

For all other latitude lines, adjacent pairs of coordinate positions are used to deﬁne a

horizontal edge of a quadrilateral, and adjacent positions along a longitude line joining

two latitude lines are used to deﬁne a vertical edge of a quadrilateral. To facilitate

processing and to ensure that the vertex positions for each surface facet are coplanar,

additional calculations can be performed to split each quadrilateral into two triangles.

The ﬁnal step in the algorithm is to store the information for vertex coordinates, polygon

edges, and surface facets in polygon tables (Section 3-15).

8-2. A polygon-mesh representation for an ellipsoid can be constructed using Eqs. 8-4 and

the sphere algorithm from the preceding exercise. The primary diﬀerences between the

two algorithms are that the parametric equations involve a radius value for each axis and

input values are needed for r

, r

, and r

, with the ellipsoid centered at the coordinate

origin. Also, coordinate positions along an ellipse are symmetric between quadrants, but

not between octants.

8-3. Assuming that the cylinder parameters are speciﬁed relative to the coordinate origin,

input for this algorithm can be the cylinder radius r and the height h above the xy plane.

Thus, the cylinder axis is along the z axis, the center position for the bottom of the

cylinder is (0, 0, 0), and the center position for the top of the cylinder is (0, 0, h). If some

other position is to be given for the cylinder, calculations for the polygon-mesh vertex

coordinates can be carried out relative to the given cylinder reference position. Or, the

cylinder can be translated and rotated to place its base in the xy plane and its axis along

the z axis.

First, compute a set of equally spaced points around the perimeter of the cylinder top.

This can be accomplished using the midpoint circle algorithm (Section 3-9) or the para-

metric circle equations 3-28. In either case, we need only calculate coordinates for the

points in the ﬁrst quadrant, and the remaining points are determined by symmetry (Fig.

3-18). These points provide the vertices for a convex polygon representation of the top of

the cylinder, and form the basis for obtaining the polygon representation for the cylinder

base. For every position (x, y, h) along the top of the cylinder, there is a corresponding

point (x, y, 0) at the base of the cylinder, which are the remaining vertex positions

needed for the polygon at the base and for the rectangles along the side of the cylinder.

The ﬁnal step in the algorithm is to store the information for vertex coordinates, polygon

edges, and surface facets in polygon tables (Section 3-15).

The number of circle subdivisions can be an additional input parameter. Or a ﬁxed

number of subdivisions, such as 8 or 10, can be used for all input cylinders.

To facilitate processing and to ensure that the vertex positions for each surface facet are

coplanar, additional calculations can be performed to convert the polygon representation

into a triangular mesh. This is accomplished for the top and bottom of the cylinder by

adding radial lines from the circle center to the vertex positions on the circle perimeter.

For each rectangular strip along the side of the cylinder, add a diagonal line. Also, the

size of the polygons along the side of the cylinder can be reduced by ﬁrst dividing each

rectangle into a number of smaller rectangles, then splitting the reduced rectangles into

triangles. Thus, another option for this algorithm is the speciﬁcation of the number of

horizontal slices to be used in creating the rectangles along the side of the cylinder.

8-4. A polygon-mesh representation for a superellipsoid can be constructed using Eqs. 8-12

and the ellipsoid algorithm (Exercise 8-2), along with the additional parameters s

and

8-5. An algorithm for generating a polygon-mesh representation for the surface of a meta-

ball object can be set up using techniques similar to those for the sphere and ellipsoid

algorithms (Exercises 8-1 and 8-2). Given values for parameters b, d, and the surface

constant for f(r), convert the object description to a parametric form involving surface

parameters u and v. Then, subdivide the surface into a number of subintervals in the

two orthogonal directions to obtain a set of polygon facets.

8-6. Input for this routine can be a data ﬁle that includes a value for the tension parameter

t, as well as a set of four control-point positions. An additional input can be either the

number of coordinate positions to plot along the curve or a step size ∆u for the parametric

parameter u.

The parametric representation 8-35 can be rewritten in standard polynomial form for

each of the x and y coordinates, and positions along the curve between the middle two

control points are calculated for parameter values u

j+1

= u

+∆u, where u

=0,u

=1,

and j =0, 1, 2,...,n. These coordinate positions could be connected with straight line

segments, or forward diﬀerences can be used to determine points along the cardinal-spline

path.

To display the curve section, the example program in Section 8-10 can be modiﬁed so

that the cardinal-spline routine replaces procedure bezier. As an option, a closed curve

as in Fig. 8-29 can be displayed using a cyclic permutation of the four control points.

Another option is to specify more that four control points so that multiple connected

cardinal-spline sections can be generated between all control points except the ﬁrst point

and the last point.

8-7. The parametric equations for a Kochanek-Bartels spline can be derived from boundary

conditions 8-36 using the same procedures discussed for cardinal splines. And the curve

50 CHAPTER 8. THREE-DIMENSIONAL OBJECT REPRESENTATIONS

can be displayed using the cardinal-spline procedures in Exercise 8-6. For this program, an

input ﬁle can include values for the tension, bias, and continuity parameters, in addition

to the control-point positions and a value for the parametric step size ∆u.

8-8. As noted in Section 8-10, a B´ezier curve generated with three distinct, noncollinear control

points is a parabola. The three blending functions are

BEZ

0,2

=(1−u)

,BEZ

1,2

=2u(1 −u),BEZ

2,2

= u

Function BEZ

0,2

has a minimum value of 0 at u =1.0 and a maximum value of 1.0 at

u = 0. Function BEZ

1,2

has its minimum value of 0 at both u = 0 and u =1.0, and

BEZ

1,2

has a maximum value of 0.5 at u =0.5. Function BEZ

2,2

has a minimum value

of0atu = 0 and a maximum value of 1.0 at u =1.0.

8-9. The ﬁve B´ezier blending functions for n = 4 are

BEZ

0,4

=(1− u)

, min = 0 at u = 1; max = 1 at u =0

BEZ

1,4

=4u(1 − u)

, min = 0 at u =0, 1; max = 27/64 at u =1/4

BEZ

2,4

=6u

(1 − u)

, min = 0 at u =0, 1; max = 3/8 at u =1/2

BEZ

3,4

=4u

(1 − u), min = 0 at u =0, 1; max = 27/64 at u =3/4

BEZ

4,4

= u

, min = 0 at u = 0; max = 1 at u =1

8-10. For this exercise, replace the set of four control-point coordinates given in procedure

displayFcn with input from a data ﬁle. Other input values could also be included in

the data ﬁle, such as the curve width, the curve color, and details for displaying the

control points and control graph. For example, the control points could be displayed in

a speciﬁed size and color, with connecting dashed lines.

8-11. For this exercise, replace the ﬁxed values for variable nCtrlPts and the control-point

coordinates given in procedure displayFcn with input from a data ﬁle that contains the

control-point positions and the number of positions. Various other data can be included

in the input, such as the number of points to be plotted along the curve path, the width

and color of the curve, and details for displaying the control points and control graph.

For example, the control points could be displayed in a speciﬁed size and color, with

connecting dashed lines. Other options include displaying the curve as set of straight-line

segments, or a set of points obtained with forward-diﬀerence calculations.

8-12. This programming exercise is a variation of the program for Exercise 8-10. Modify the

programming example in Section 8-10 so that the OpenGL routines from the example

in Section 8-18 are used to generate and display the B´ezier curve, instead of the explicit

calculations for the blending functions and curve coordinates.

8-13. Expand the program from the preceding exercise so that the control points are speciﬁed

in three-dimensional Cartesian coordinates and a three-dimensional B´ezier curve is gener-

ated. Viewing routines for displaying the curve can be taken from previous programming

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