Review: Math (Ch 4)

2/7/2001 Hofstra University – CSC290B 1

Review: Math (Ch 4)


General Purpose Graphics

API for use in high level languages Appl.program ------- Graph.API ------- Graph.hardware

Primitives and Attributes Geometric and modeling transformations Viewing transformations Hierarchical modeling and manipulation Raster (buffer) oriented transformations Control functions


Fundamental Operations

Transformation – changes in coordinate systems through matrix operations

Clipping – adjusting our view of the world Projection – from 3D to 2D Rasterization – pixels in the frame buffer


Geometric Transformations

• Work in Euclidean affine space:

vectors, + , multiplication with scalars, 0 linear combinations, linear independence, basis linear maps (transformations), change of basis

points, point+vector, point –point affine combinations, frame

inner product, distances, angles, unit vector, perpendicular

orthonormal frame, change of frames, affine maps


Review: Affine Transformations change objects so programmer

can:

Model objects in simple configurations

Make multiple copies of objects Animate objects, move or deform

over time


Vectors

Directed line segment that connects points

Does not have a fixed position


Vectors

B = 2A

A

C = -A

B

A

C = A + B

head-to-tail rule

Linear independence, basis, linear maps


Geometric ADTs

Scalars, Points and Vectors are members of mathematical sets

Abstract spaces for representing and manipulating these sets of objects

Linear Vector Space – scalars & vectors

Affine Space – adds the point Euclidean Space – add concept of

distance and angle


Geometric ADTs Operations that relate points and

vectors Subtraction of two points yields a

vector:v = P – Q

Point-vector addition yields a point:

P = v + Q


Geometric ADTs: Lines Consider all points of the form

P () = P0 + dwhere P0 is an arbitrary point, d is an arbitrary vector, and is a scalar

P () is a point for any value of Called the parametric form of the line Generate points on the line by

varying the parameter For non-negative values, we get

a ray emanating from P0 in the direction of d

d

P 0

P (a )


Geometric ADTs: Affine Sums

For any point Q, vector v, and positive scalar ,

P = Q + vdescribes all points on the line from Q in the direction of v

Affine addition is characterized by the addition of two arbitrary points and multiplication of a point by a scalar,

P = R + Q where 1 + 2 = 1

v

Q

P () = 1R

= 0


Geometric ADTs: Dot Product

We are concerned about the orientation between two vectors

Dot (Inner) Product of vectors u and v is written

u v If u v = 0, then u and v are orthogonal In Euclidean space, the square of the

magnitude of a vector is |u|2 = u u

The angle between two vectors is given byvuvu=cos


Geometric ADTs: Cross Product

Cross Product – is determined by a third vector n that is orthogonal to two other vectors v and u. It is denoted asn = u v


3D Primitives

Curves Surfaces Volume Objects


3D Primitves

Objects With Good Characteristics Described by their surfaces;

thought to be hollow Specified through a set of vertices

in 3D Composed of or approximated by

flat convex polygons


Coordinate Systems

Given a frame (P0, v1, v2, v3), a vector w:

And in the same frame a point P:

3322110 vvvPP =

332211 vvvw =


Changes of Basis

How do we represent the vector if we change the basis?

Suppose the {v1,v2,v3} and {u1,u2,u3} are two bases.

Basis vector in second set can be represented in terms of the first basis


Change of Basis in Linear Space

3332321313

3232221212

3132121111

vvvu

vvvu

vvvu

===

=

333231

232221

131111

M

U = MV


Change of Basis

=

31

21

11

a

=

31

21

11

b

=

333231

232221

131111

M

bMa T=

In V basis, the vector is represented by a, and in U by b

U=MV, b’U=b’MV=a’V, a’=b’M, a=M’b

aMAab T 1)( ==


Homogeneous CoordinatesRepresent a point at (x,y,z) wrt a 3D frame determined by (P0, v1, v2, v3 ) with a column matrix

Where x,y,z are the components of the basis vectors for this point, so that

and represent a vector in the same form

=

z

y

x

p

3322110 vzvyvxPP =

=

3

2

1

w


Homogeneous Coordinates Maintain distinction between

points and vectors Use four dimensional column

matrices to represent both points and vectors in three dimensions

If we assume a frame specified by (v1, v2, v3, P0), then any point P can be written

0332211 PvvvP =


Homogeneous Coordinates …and if we define multiplication of

a point by the scalars 0 and 1 as0 ·P = 0,1 . P = P,

then we can use a matrix product to express this relation formally as

=

0

3

2

1

321 1

P

v

v

v

P

homogeneous-coordinate representation


Homogeneous Coordinates

We carry out operations on points and vectors using their homogeneous-coordinate representation and ordinary matrix algebra

=

13

2

1

p

=

03

2

1

a

Point Vector


Homogeneous Coordinates

=

0

3

2

1

0

3

2

1

P

v

v

v

Q

u

u

u

M

=

1

0

0

0

434241

333231

232221

131211

M

bMa T=

All affine transformations can be represented as matrix multiplications in homogeneous coordinates

two frames change of framesmatrix representation

aMAab T 1)( ==


Frames In OpenGL We use two frames: the camera frame and the

world frame We regard the camera frame as fixed The model-view matrix positions the world frame

relative to the camera frame Perform homogeneous coordinate

transformations on object in other frames to define new frames relative to the camera frame

Model-view Matrix: translate along z, to separate the two frames, so object could be in camera’s field of view


Frames In OpenGL


Modeling

Select modeling (local) coordiante system best suited for modeling the object

Define objects in terms of vertices Pass vertices through a number of

transformations using homogeneous coordinates


Affine Transformations A map f () is linear iff, for any scalars and , and any

vectors p and q,f(p+ q) = f(p)+ f(p)

Affine maps preserve affine combinations of points f(P+ Q) = f(P)+ f(Q), where + =1 are linear in homogenious coordinates ( i.e. represented by

4x4 matrix, last row (0,0,0,1), act as matrix multiplications). the transf. of affine combinations of points is same affine

combination of the transformed points can be represented as a composition of translations,

rotations, and scalings


Transformations in Homogeneous Coordinates

Each affine transformation is represented by a 4 x 4 matrix of the form

=

100034333231

24232221

14131211

M


Graphics Pipeline

Pipeline endpoints (in homogeneous coordinates) through affine transformations to generate the rastered image


Concatenation of Transformations

We can multiply together sequences of transformations – concatenating

Works well with pipeline architecture e.g., three successive transformations

on a point p creates a new point qq = CBAp


Concatenation of Transformations

If we have a lot of points to transform, then we can calculate

M = CBAand then we use this matrix on each point

q = Mp


Translation

Translation is an operation that displace points by a fixed distance in a given direction

Only need to specify a displacement vector d

Transformed points are given by P = P + d


Translation


Rotation about a pivot point P in 2D

R(P;u)=T(-P)R(O;u)T(P)


Rigid Body Transformation•Rotation and translation are rigid-body transformations•No combination of transformations can alter the shape of an object

Non-rigid-body transformations


Scaling

non-uniform

uniform


Rotation About Axis Parrallel to z-axis

Move the cube to the originApply Rz()Move back to original position

)()()( fzf pTRpTM =


Instance Transformation

objectprototype

instance


Instance Transformation


Current Transformation Matrix

Current Transformation Matrix (CTM) – matrix that is applied to any vertex that is defined subsequent to setting the CTM

Changing the CTM, alters the state of the system

4x4 matrix that can be altered by a set of functions provided by the graphics package

Common to most systems. Part of the pipeline If p is a vertex, the pipeline produces Cp



Let C denote the CTM. It is set to the 4x4 identity matrix, initially. The symbol denotes replacement

Initialization operation:C I

Three transformations about the fixed point of the origin: translation, rotation, scaling

Set: Postmultiplication:C T C CTC R C CRC S C CS



The matrix the is applied to all primitives is the product of the model-view matrix (GL_MODELVIEW) and the projection matrix (GL_PROJECTION)

The CTM is the product of these matrices!



The matrix that is applied to all primitives is the product of the model-view matrix (GL_MODELVIEW) and the projection matrix (GL_PROJECTION)

The CTM is the product of these matrices

We select the desired matrix of each with glMatrixMode then alter the selected matrix through postmultiplication using:

glLoadIdentity( );glRotatef(angle, vx, vy, vz);glTranslatef(dx, dy, dz);glScalef(sx, sy, sz);


Order of Transformations Transformation specified most

recently is the one applied firstglMatrixModel(GL_MODELVIEW)glLoadIdentity( );glTranslatef(4.0, 5.0, 6.0);glRotatef(45.0, 1.0, 2.0, 3.0);glTranslatef(-4.0, -5.0, -6.0);


Summary: Affine Maps

Composition of translations, rotations, scalings

Preserve affine combination of points

If object is modeled as affine combination from vertices, under affine map, the transformed object is the same affine combination of the transformed vertices.


Summary: Affine Maps

Liner maps in homogeneous coordinates, represented by matrices

Composition of affine maps is represented by a matrix which is product of the matrices of the individual maps.


Fundamental Operations

Transformation – changes in coordinate systems through matrix operations

Clipping – adjusting our view of the world Projection – from 3D to 2D Rasterization – pixels in the frame buffer

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Review: Math (Ch 4)

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