15%20March%202006

1
Chapter 4

• 15 March 2006

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2
Agenda

• Chapter 4 Math for Computer Graphics
• GLUT solids

3
Transformations

• 45-degree counterclockwise rotation about the
  origin around the z-axis
• a translation down the x-axis

4
Order of transformations
glMatrixMode(GL_MODELVIEW) glLoadIdentity() glMu
ltMatrixf(N) / apply
transformation N / glMultMatrixf(M)
/ apply transformation M / glMultMatrixf(L)
/ apply transformation L
/ glBegin(GL_POINTS) glVertex3f(v)
/ draw transformed vertex v / glEnd()

• transformed vertex is NMLv

5
Translation

• void glTranslatefd (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

6
Rotation

• void glRotatefd(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.

7
Scale

• void glScalefd (TYPEx, TYPE y, TYPEz)
• Multiplies the current matrix by a matrix that
  stretches, shrinks, or reflects an object along
  the axes.

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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

9
VectorsVector and Vector Algebra
1 2 3 2 3 5 3 4 7
10
Matrices

• Rectangular array of numbers
• Addition
• QuickMath

11
Matrices

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

12
Matrices

• Multiplication

1 0 0 0 0 1 0 0 x 0 0 0 0 0
0 1/k 1
Cos a 0 sin a 0 0 1 0 m -sin a
0 cos a n 0 0 0 1
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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
• s1

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Matrix multiplication

• m
• cij ?ais bsj
• s1

c11 c12 c13 c14 c21 c22 c23
c24 c31 c32 c33 c34 c41 c42 c43
c44
1 0 0 0 0 1 0 0 x 0 0 0 0 0
0 1/k 1
Cos a 0 sin a 0 0 1 0 m -sin a
0 cos a n 0 0 0 1
a
b
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2D Transformations

• Translation Pf T P
• xf xo dx
• yf yo dy
• Rotation Pf R P
• xf xo cos? - yo sin?
• yf xo sin? yo cos?
• Scale Pf S P
• xf sx xo
• yf sy yo

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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

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Homogeneous Coordinates

• 1 0 dx x
• 0 1 dy y
• 0 0 1 1

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Homogeneous Coordinates

• sx 0 0 x
• 0 sy 0 y
• 0 0 1 1

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Homogeneous Coordinates

• Cos? -sin? 0 x
• sin? cos? 0 y
• 0 0 1 1

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Combining 2D Transformations

• Rotate a house about the origin
• Rotate the house about one of its corners
• Translate so that a corner of the house is at the
  origin
• Rotate the house about the origin
• Translate so that the corner returns to its
  original position

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GLUT Solids

• Sphere
• Cube
• Cone
• Torus
• Dodecahedron
• Octahedron
• Tetrahedron
• Icosahedron
• Teapot

22
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).

23
glutSolidCube and glutWireCube

• void glutSolidCube(GLdouble size)
• size length of sides

24
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.

25
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.

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glutSolidDodecahedron and glutWireDodecahedron

• void glutSolidDodecahedron(void)

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glutSolidOctahedron and glutWireOctahedron .

• void glutSolidOctahedron(void)

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glutSolidTetrahedron and glutWireTetrahedron

• void glutSolidTetrahedron(void)

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glutSolidIcosahedron and glutWireIcosahedron

• void glutSolidIcosahedron(void)

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glutSolidTeapot and glutWireTeapot

• void glutSolidTeapot(GLdouble size)
• size - Relative size of the teapot.

31
Homework next week.

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