Geometric Objects and Transformations

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Geometric Objects and Transformations

• Chapter 4

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Points, Scalars and Vectors

• Points - position in space
• Scalars - real numbers, complex numbers obey a
  set of rules that are abstractions of arithmetic
  store values such as distance
• Vectors - directed line segment

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Vector Arithmetic

• B 2A
• C A B
• Head to tail rule
• E -A
• Inverse
• Zero vector E A

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Point Arithmetic

• Cant add p1 p2 p3
• Cant multiply p1 2 p2
• What can you do to produce a 2nd point from a
  1st?
• p1 V p2
• V p2 - p1

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Important Vector Concepts

• Normal Vector
• A vector at a right angle to a surface.
• Graphics Usage of Normal Vector
• Used to figure out similarity of direction which
  is necessary for lighting
• More precisely, how much light should fall on a
  surface is calculated by the similarity of the
  surface normal and the vector between the light
  source and the surface.
• Similarity of direction vector dot product

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Vector Dot Product

• For vectors of length 1 (unit vectors), the dot
  product is the length of the projection of one
  vector onto the other.
• If the dot product
• 1 the vectors point in the same direction
• 0 the vectors are at right angles
• -1 the vectors point in opposite directions

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Vector Cross Product

• The cross product of two vectors A and B is
  another vector at right angles to the plane
  created by A and B

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Defining a Coordinate Space

• Need to know
• The origin (or displacement vector)
• The basis vectors - The direction and distance
  for 1 movement along each axis
• This definition is relative
• To plot a point
• Begin at origin
• Travel along the x basis vector direction
  scaled by x coord, then along the y basis vector
  scaled by the y coord, then finally along the z
  basis vector scaled by the z coord.

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Transformations

• Prior to rendering view, locate and orient
• eye / camera position
• 3D geometry
• Manage the matrices
• including the matrix stack
• Combine (composite) transformations

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Camera Analogy
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Stages of Vertex Transformation
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Transformations

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

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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

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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

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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.

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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
1 2 3 2 3 5 3 4 7
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Matrices

• Rectangular array of numbers
• A vector in 3 space is a n x 1 matrix or
  column vector.
• 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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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
• 1 x 0 y dx 1
• 0 x 1 y dy 1
• 0 x 0 y 1 1

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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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Homogeneous Coordinates

• 1 0 0 x x
• 0 1 0 y y
• 0 0 1 1 1
• Identity Maxtrix x point p point p

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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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OpenGL Buffers

• Color
• can be divided into front and back for double
  buffering
• Alpha
• Depth
• Stencil
• Accumulation

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Double Buffering
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Animating Using Double Buffering

• Request a double buffered color buffer
• glutInitDisplayMode (GLUT_RGB GLUT_DOUBLE)
• Clear color buffer
• glClear(GL_COLOR_BUFFER_BIT)
• Render Scene
• Request swap of front and back buffers
• glutSwapBuffers()
• Repeat steps 2-4 for animation.

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Depth Buffering
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3D Coords --gt Raster coords

• Transformations
• Clipping
• Viewport transformation.

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GLUT Solids

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

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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).

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glutSolidCube and glutWireCube

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

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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.

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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.

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Homework

• Project part 2 due 2/19
• Turn in a program that, at a minimum, draws your
  initial scene.