Chapter 4

15 March 2006

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Agenda

Chapter 4 – Math for Computer Graphics

GLUT solids

Transformations

45-degree counterclockwise rotation about the origin around the z-axis

a translation down the x-axis

Order of transformations

transformed vertex is NMLv

glMatrixMode(GL_MODELVIEW);glLoadIdentity();glMultMatrixf(N); /* apply transformation N */glMultMatrixf(M); /* apply transformation M */glMultMatrixf(L); /* apply transformation L */glBegin(GL_POINTS);glVertex3f(v); /* draw transformed vertex v */glEnd();

Translation

void glTranslate{fd} (TYPE x, TYPE y, TYPE z);

Multiplies the current matrix by a matrix that moves (translates) an object by the given x, y, and z values

Rotation

void glRotate{fd}(TYPE angle, TYPE x, TYPE y, TYPE z);

Multiplies the current matrix by a matrix that rotates an object in a counterclockwise direction about the ray from the origin through the point (x, y, z). The angle parameter specifies the angle of rotation in degrees.

Scale

void glScale{fd} (TYPEx, TYPE y, TYPEz);

Multiplies the current matrix by a matrix that stretches, shrinks, or reflects an object along the axes.

Vectors

N tuple of real numbers (n = 2 for 2D, n = 3 for 3D)

directed line segment example

velocity vector (speed and direction) operations

addition multiplication by a scalar dot product

VectorsVector and Vector Algebra

1 2 32 + 3 = 53 4 7

Matrices

Rectangular array of numbers Addition QuickMath

Matrices

A vector in 3 space is a n x 1 matrix or column vector.

131

Matrices

Multiplication

1 0 0 00 1 0 0 x 0 0 0 00 0 1/k 1

Cos α 0 sin α 0 0 1 0 m-sin α 0 cos α n 0 0 0 1

Matrix multiplication

A is an n x m matrix with entries aij

B is an m x p matrix with entries bij

AB is an n x p matrix with entries cij

m

cij = ais bsj

s=1

Matrix multiplication

m

cij = ais bsj

s=1

1 0 0 00 1 0 0 x 0 0 0 00 0 1/k 1

Cos α 0 sin α 0 0 1 0 m-sin α 0 cos α n 0 0 0 1

a b

c11 c12 c13 c14

c21 c22 c23 c24

c31 c32 c33 c34

c41 c42 c43 c44

2D Transformations

Translation: Pf = T + Pxf = xo + dxyf = yo + dy

Rotation: Pf = R · Pxf = xo * cos - yo *sin yf = xo * sin + yo *cos

Scale: Pf = S · Pxf = sx * xo

yf = sy * yo

Homogeneous Coordinates

Want to treat all transforms in a consistent way so they can be combined easily

Developed in geometry (‘46 in Cambridge) and applied to graphics

Add a third coordinate to a point (x, y, W) (x1, y1, W1) is the same point as (x2, y2,

W2) if one is a multiple of another Homogenize a point by dividing by W

Homogeneous Coordinates

1 0 dx x0 1 dy · y 0 0 1 1

Homogeneous Coordinates

sx 0 0 x 0 sy 0 · y 0 0 1 1

Homogeneous Coordinates

Cos -sin 0 x sin cos 0 · y 0 0 1 1

Combining 2D Transformations Rotate a house about the origin Rotate the house about one of its

cornersTranslate so that a corner of the

house is at the originRotate the house about the originTranslate so that the corner returns to

its original position

GLUT Solids

Sphere Cube Cone Torus Dodecahedron Octahedron Tetrahedron Icosahedron Teapot

glutSolidSphere and glutWireSphere

void glutSolidSphere(GLdouble radius, GLint slices, GLint stacks);

radius - The radius of the sphere. slices - The number of subdivisions

around the Z axis (similar to lines of longitude).

stacks - The number of subdivisions along the Z axis (similar to lines of latitude).

glutSolidCube and glutWireCube

void glutSolidCube(GLdouble size);

size – length of sides

glutSolidCone and glutWireCone void glutSolidCone(GLdouble base, GLdouble height, GLint slices, GLint stacks); base - The radius of the base of the cone. height - The height of the cone. slices - The number of subdivisions around

the Z axis. stacks - The number of subdivisions along

the Z axis.

glutSolidTorus and glutWireTorus void glutSolidTorus(GLdouble innerRadius,GLdouble outerRadius, GLint nsides, GLint rings); innerRadius - Inner radius of the torus. outerRadius - Outer radius of the torus. nsides - Number of sides for each radial

section. rings - Number of radial divisions for the

torus.

glutSolidDodecahedron and glutWireDodecahedron

void glutSolidDodecahedron(void);

glutSolidOctahedron and glutWireOctahedron .

void glutSolidOctahedron(void);

glutSolidTetrahedron and glutWireTetrahedron

void glutSolidTetrahedron(void);

glutSolidIcosahedron and glutWireIcosahedron

void glutSolidIcosahedron(void);

glutSolidTeapot and glutWireTeapot

void glutSolidTeapot(GLdouble size);

size - Relative size of the teapot.

Homework next week.

Study for Test on Chapters 1-4, 02/15/05