Geometric Objects and Transformations
description
Transcript of Geometric Objects and Transformations
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Geometric Objects and Transformations
Coordinate systems and framesWorking with representation
Object transformation
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Introduction
• Mathematical of Object– Euclidean vector spaces
• Vector space with measure of size– Independent of coordinate system
• Parametric form system
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Scalars, Points, and Vectors
• Geometric object description in space– By length, angle– With 3 fundamental types scalars, points and
vector
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The geometric View
• Point : a location in the space– Mathematical for point
• Neither a size nor shape– Properties
• location– What for :
• Specify an object
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Scalars
• Quantity of object or relation objects• Ex. Distance between object• Specify with real / complex number• Useful rule for scalar
– communitivity and Associativity in multiplicity, additivity
– Ex. a+b = b+a, (a+b)+c = a+(b+c)
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Vector
• Quantity with direction and magnitude ex. velocity, force– Does not have a fixed position in space– Synonymously to line segment
• Computer graphics often connect points with directed line segment– Line segment: a segment of line which
has both magnitude and direction
Directed line segment that connects points
Identical vectors
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Vector properties• Its lengths changed by real number• B = 2A
– B is double in size to vector A with the same direction• Vector combining: (addition)
– Use head to tail combining, the result is the sum of the vector– Any vector in space is able to do addition, independent of its location– Scalar-vector addition make sense: ex. A + 2B – 3C
• Inverse vector:– The vector that has opposite direction to the original vector
Parallel line segments
Addition of line segments Inverse vectors
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Vector operation• Scalar multiplying
– Result change in length (magnitude) • Point-vector addition
– Result change in displacement to the new point position
• Point-point subtraction– Result is a vector between 2 points
• Note: Some expression involving scalars, vectors and
points make sense ex. P+3v, or P-2Q+3v while P+3Q-v do not
Point-vector addition
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Coordinate-free geometry
• For graphics system let the object relate to each other but independent of coordinate system
• Let object relate to an arbitrary location and orientation of the original axis
Object and coordinate system
Object without coordinate system
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The mathematical view: Vector and Affine Spaces
• Scalar operation– Addition , multiplication
• If operation obey closure, associativity, commutivity and inverse properties, the element form a scalar field
– Ex. field . Real number, complex number, rational function
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Vector space
• 2 distinct type of entities in vector space– Scalar and vector
• Scalar-vector multiplication– Vector and vector
• Vector – vector addition
• Euclidean space– Extension of vector space– Add a measure of size or distance to define object– Ex. Length of line segment
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Affine space (space of transformation)
• Extension of vector space– Include point to vector space– Have vector-point addition and point-point
operation
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Computer Science View• Prefer to see object as
abstract data type (ADT)• Operations and data are
defined independently• Fundamental to modern
computer science• Like C++ language features :
class and overloading
Computational point of view declaration (Independent of data type declaration)
Operation that independent of data type
vector u,v;point p,q;scalar a, b;
q = p+a*v;
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Geometric ADT
• Learn how to perform geometric operation and forming geometric object
• Let Greek letter a, b, g,... : scalars
Upper-case letter P, Q, R,… : points
Bold lower-case letters u, v, w,… : vector
• The operation of vector-scalar multiplication
• Subtraction of two points, P and Q -> vector v
• Add vector with point get point
• Vector-vector operation can be in point form
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Point-point subtraction
Use of the head-to-tail rule(a) For vectors, (b)For points
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Lines
• General term definition• Called parametric form of line• Point generating by varying alpha• Line is infinity in length• One we see just line segment
Parametric line equation
Line in an affine space
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Affine sumIf P is a point on line
P=Q+avv = R – Q;
ThusP = Q + a(R-Q)
= aR + (1-a)QLet
a = a1 and (1-a) = a2
Thus a1 + a2 = 1
Then P=a1R+a2Q
Points on line can be found between point Q and P(a)
Affine line addition
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Convexity
• Convex object – Any point that lying on
line segment connect any two points in the object also in the object
• Affine sum definition
...1 1 2 2 n nP = P P Pa a a When
... 11 2 n =a a a
Objects defined by n points P1, P2,…,Pn. Consider the form
Line segment that connects two points
• An object is convex iff for any two points in the object all points on the line segment between these points are also in the object
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P
Q Q
P
convexnot convex
Convexity
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is called the convex hull of the set of pointsconvex hull includes all line segments connecting pairs of points i[P1,P2,…,Pn]The notion of convexity is extreamly important in the design of curves and surfaces;
Convex hull
Convex hull
The set of points formed by the affine sum of n points, under the additional restriction
iα 0, i = 1,2,...,n
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Dot and Cross product
• Dot (inner product)– u.v: result is magnitude of 2 vectors– If u.v = 0, u and v are orthogonal vector– Unit product is
Dot product and projection
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Cross product• Result is vector• Forming from right hand coordinate system
Note: right-handed coordinate systemu points in the direction of the thumb v points in the direction of the index fingern points in the direction of the middle finger
Cross product
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Plane• Infinite flat area• Direct extension of parametric line• Define with 3 non-co-linear points• Suppose P, Q and R are points in plane• The plane equation
and
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The plane can have vector (normal vector) formed from u and v
Let
n : plane vector
Thus
General plane equation
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3D primitives
Object are not lying on plane
Curves in three dimension Surfaces in three dimensions Volumetric objects
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2 problems when incorporate in 3D
• Complex mathematic– Not all object that has 3D efficient
implementation• Approximated method may be used• 3 features characteristic to describe object
– Hollow– Vertices– Flat convex polygon composition
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• Represents Vector in 3D space with 3 basis vector
thus
Coordinate system and frame
Vector derived from three basis vectors
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Point and vector representation in affine space• It is not enough to use only vector to represent
point in space• Frame: fix point (origin) and basis vector
– Vector: represent with 3 basis– Point: fix point and 3 basis
Coordinate system (a) with vector emerging from a common point, (b) with vector moved
A Dangerous representation of vector
Representations and N-tuples• Any vector v representation
Where
• In column matrix form
• Known as Euclidean :
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Coordinate System ChangingWorld /User frame Camera frame
– Done by MODELVIEW matrix– Use 2 basis set of vector and – Representation of by
Let
and
represent matrix from to
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Let
Equivalent to
Where
Assume : the representation of with respect to
or
Where
Then using our representation of the second basis in terms of the first, we find that
Thus
The matrix (MT)-1 takes us from a to b:
We can transfer coefficient matrix from one to the other
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• This changing let us to work with different coordinate system but origin unchanged
• Able to use them to represent rotation and scaling but not translation
Rotation and scaling of a basis Translation of a basis
2 examples 2 different frames
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Example: Suppose that we have a vector whose representation in some basis is
1 a 2
3
We can denote the three basis vectors and , , and hence, = +2 +3Now suppose that we want to
1 2 3
1 2 3
w
=
v v vw v v v
1 1
2 1 2
3 1 2 3
make a new basis from the three vectors , , and The matrix M is
1 0 0 M= 1 1 0
1 1 1
1 2 3v v vu vu v vu v v v
The matrix that converts a representation in , and to one in which the basis vector are and is
In the new system, the representation of w is
That is
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Homogeneous Coordinate3 basis vector is not enough to make point and vector differentFor any point:
In the frame specified by (v1, v2, v3, P0), any point P can be written uniquely as
The we can express this relation formally, using a matrix product, as
The yellow matrix called homogeneous-coordinate representation of point P
In the same frame, any vector w can be written
w can be represented by the column matrix
If and , can be expressed as
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Can be written in the form
M : the matrix representation of the change of frames.
Suppose a and b are the homogeneous-coordinate representations, then
Hence,
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When we work with representations, as is usually the case, we are interested in , which is of the form
Only 12 coefficients
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Advantage of using homogeneous representationo use 4D matrix instead of 3D matrix
o less calculationo modern hardware support the representation
o parallelism for high speed calculation
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Example
1 1
2 1 2
3 1 2 3
If we start with the basic vector , and and convert to a basis determined by the same , and then the three equations are the same
,
The reference point does not c
1 2 3
1 2 3
v v vu u u
u vu v vu v v v
hange, so we add the equation =
Thus, the matrices in which we are interested are the matrix1 0 0 01 1 0 0
M=1 1 1 00 0 0 1
its transpose, and their inverses.
0 0Q P
Suppose move the point to (1, 2, 3, 1)
with displacement vector
And move from point P0 to Q0 then
The matrix becomes
Its inverse is
We can move back and forth between representation
Thus
to
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Original vector
b= (MT)-
1a<-
The origin in the new system is represented as 12
a30
11
b30
is transformed to
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Working with Representation• Represent object from a frame to another such as
world frame to camera frame• Form of representation
a = Cb ; a, b object in two representation frame• Object:
– 3 vector, u, v, n, and 1 new frame origin, p -> (u, v, n, p)– 4 entities -> 4-tuples -> R4
• The solution is inverse matrix form of C, let’s say DD = C-1
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1 1 1 1
2 2 2 2
3 3 3 3
11 1 1 1
1 2 2 2 2
3 3 3 3
DI D
0 0 0 1or
C
0 0 0 1
u v n pu v n p
u v n pu v n p
u v n pu v n p
u v n pu v n p
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Frames in OpenGL• 2 frames:
– Camera: regard as fix– World:
• The model-view matrix position related to camera• Convert homogeneous coordinates representation of object to camera
frame
• OpenGL provided matrix stack for store model view matrices or frames
• By default camera and world have the same origin• If moving world frame distance d from camera the
model-view matrix would be
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Camera and world frames
1 0 0 00 1 0 0
A=0 0 10 0 0 1
-d
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Suppose camera at point (1, 0, 1, 1)
• World frame center point of camera
• Representation of world frame for camera
• Camera orientation: up or down
• Forming orthogonal product for determining v let’s say u
Camera at (1,0,1) pointing toward the origin
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The model view matrix M
Result: Original origin is 1 unit in the n direction from the origin in the camera frame which is the point (0, 0, 1, 1)
OpenGL model view matrixOpenGL set a model-view matrix by send an array of 16 elements to glLoadMatrixWe use this for transformation like rotation, translation and scales etc.
1
T 1
1 10 01 0 1 1 2 20 1 0 0 0 1 0 0
(M )1 0 1 1 1 10 1
2 20 0 0 10 0 0 1
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Modeling a Color Cube• A number of distinct task that we
must perform to generate the image– Modeling– Converting to the camera frame– Clipping– Projecting– Removing hidden surfaces– Rasterizing One frame of cube animation
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Modeling a Cube• Model as 6 planes intersection or six polygons as cube facets• Ex. of cube definition
// object may defined as
void polygon(int a, int b, int c , int d) {/* draw a polygon via list of vertices */ glBegin(GL_POLYGON);
glVertex3fv(vertices[a]);glVertex3fv(vertices[b]);glVertex3fv(vertices[c]);glVertex3fv(vertices[d]);
glEnd();}
GLfloat vertices[8][3] = {{-1.0,-1.0,-1.0}, {1.0,-1.0,-1.0}, {1.0,1.0,-1.0}, {-1.0,1.0,-1.0}, {-1.0,-1.0,1.0}, {1.0,-1.0,1.0}, {1.0,1.0,1.0}, {-1.0,1.0,1.0}
};// ortypedef point3[3];
// then may define aspoint3 vertices[8] = {
{-1.0,-1.0,-1.0}, {1.0,-1.0,-1.0}, {1.0,1.0,-1.0}, {-1.0,1.0,-1.0}, {-1.0,-1.0,1.0}, {1.0,-1.0,1.0}, {1.0,1.0,1.0}, {-1.0,1.0,1.0}
};
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Inward and outward pointing faces
• Be careful about the order of vertices
• facing outward: vertices order is 0, 3, 2, 1 etc., obey right hand rule
Traversal of the edges of a polygon
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Data Structure for Object Representation
• Topology of Cube description– Use glBegin(GL_POLYGON); six times– Use glBegin(GL_QUADS); follow by 24
vertices• Think as polyhedron
– Vertex shared by 3 surfaces– Each pair of vertices define edges– Each edge is shared by two faces
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Vertex-list representation of a cube
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The color cube• Color to vertex list -> color 6 faces• Define function “quad” for drawing quadrilateral polygon• Next define 6 faces, be careful about define outwarding
typedef GLfloat point3[3];
point3 vertices[8] = {{-1.0,-1.0, 1.0},{-1.0, 1.0, 1.0}, {1.0,1.0, 1.0}, {1.0,-1.0, 1.0}, {-1.0,-1.0,-1.0}, {1.0,-1.0,-1.0}, {1.0,1.0,-1.0},
{-1.0,1.0,-1.0}};GLfloat colors[8][3] = {{0.0,0.0,0.0},{1.0,0.0,0.0}, {1.0,1.0,0.0}, {0.0,1.0,0.0},
{0.0,0.0,1.0}, {1.0,0.0,1.0}, {1.0,1.0,1.0}, {0.0,1.0,1.0}};
void quad(int a, int b, int c , int d) {glBegin(GL_QUADS); glColor3fv(colors[a]); glVertex3fv(vertices[a]); glColor3fv(colors[b]); glVertex3fv(vertices[b]);
glColor3fv(colors[c]); glVertex3fv(vertices[c]); glColor3fv(colors[d]); glVertex3fv(vertices[d]); glEnd();
}void colorcube() { quad(0, 3, 2, 1); quad(2, 3, 7, 6); quad(0, 4, 7, 3); quad(1, 2, 6, 5); quad(4, 5, 6, 7); quad(0, 1, 5, 4);}
Color the cube surface
•
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Projection of polygon
Bilinear interpolation
Vertex Array• Use encapsulation, data structure & method together
– Few function call• 3 step using vertex array
– Enable functionality of vertex array: part of initialization– Tell OpenGL where and in what format the array are: part of
initialization– Render the object :part of display call back
• 6 different type of array– Vertex, color, color index, normal texture coordinate and edge flag– Enabling the arrays by
glEnableClientState(GL_COLOR_ARRAY);glEnableClientState(GL_VERTEX_ARRAY);
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// The arrays are the same as before and can be set up as globals:
GLfloat vertices[] = {{-1,-1,-1},{1,-1,1},{1,1,-1},{-1,1,-1},{-1,-1,1}, {1,-1,1}, {1,1,1}, {-1,1,1}};
GLfloat colors[] = {{0,0,0},{1,0,0},{1,1,0},{0,1,0},{0,0,1},{1,0,1}, {1,1,1},{0,1,1}};
// Next identify where the arrays are by
glVertexPointer(3, GL_FLOAT, 0, vertices);
glColorPointer(3, GL_FLOAT, 0, colors);
// Define the array to hold the 24 order of vertex indices for 6 faces
GLubyte cubeIndices[24] = {0,3,2,1, 2,3,7,6, 0,4,7,3, 1,2,6,5, 4,5,6,7, 0,1,5,4};
glDrawElements(type, n, format, pointer);
for (i=0; i<6; i++)
glDrawElement(GL_POLYGON, 4, GL_UNSIGNED_BYTE, &cubeIndex[4*i]};
// Do better by seeing each face as quadrilateral polygon
glDrawElements(GL_QUADS, 24, GL_UNSIGNED_BYTE, cubeIndices);
// GL_QUADS starts a new quadrilateral after each four vertices
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Affine transformation
Transformation:A function that takes a point (or vector) and maps that points (or vector) in to another point (or vector)Point transform
Vector transform
Transformation
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Transformation form of 4D matrix
Transformation function must be linearity function
For vector has 12 degree of freedom for point and vector
f p q f p f qa b a b
v Au
11 12 13 14
21 22 23 24
31 32 33 34
A
0 0 0 1
a a a aa a a aa a a a
1
2
3
u
0
aaa
1
2
3
1
v
bbb
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Rigid body transformation
• Shape doesn’t change• Only change in position and orientation• They are: Translation and rotation.
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Translation
• For all point P of the object– No reference point to frame or representation– 3 degree of freedom
Translation. (a) Object in original position. (b) Object translated
Is an operation that displaces points by a fixed distance in a given directionTo specify a translation, just specify a displacement vector d thus
P P d
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Rotation
• Need axis and angle as input parameter
2D Point rotation θ radian around origin (Z axis) Two-dimensional rotation
x = r.cosf ; y = r.sinf
x’ = r.cos(q f= r.cosf.cosq r.sinf.sinq= x.cosq – y.sinq
y’ = r.sin(qf)= r.cosf.sinq +r.sinf.cosq = x.sinq+y.cosq
These equations can be written in matrix form ascos sin
sin cos
x xy y
q qq q
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3 features for rotation(3 degrees of freedom)
• Around fixed point (origin)• Direction ccw is positive• A line
Rotation arount a fixed point.
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3D rotation• Must define 3 input parameter
– Fixed Point (Pf)– Rotation angle (θ)– A line or vector (rotation axis)
• 3 degrees of freedom• 2 angle necessary specified
orientation of vector• 1 angle for amount of rotation
Three-dimensional rotation
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Scaling• Non rigid-body transformation• 2 type of scaling
– Uniform: scale in all direction -> bigger, smaller– Nonuniform: scale in single direction
• Use in modeling and scaling
Non rigid body transformations
Uniform and nonuniform scaling
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Scaling: input parameter
• 3 degrees of freedom– Point (Pf) – Direction (v)– Scale factor (a)
• a > 1 : get bigger• 0 £ a < 1 : smaller• a < 0 : reflected
Reflection
Effect of scale factor
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Transformation in homogeneous coordinateMost graphics APIs force us to work within some reference system. Although we can alter this reference system – usually a frame – we cannot work with high-level representations, such as the expression.
Q = P + αv.
Instead, we work with representations in homogeneous coordinates, and with expressions such as
q = p + αv.
Within a frame, each affine transformation is represented by a 4x4 matrix of the form
TranslationIf we move the point p to p by displacing by a distance d then
p = p+dDefine
p = , p= , d=
1 1 0 may be written for each of components
x
y
z
x xy yz z
It
aaa
Let T is a translation matrix, we may written in matrix form as: And
1 0 00 1 0
T=0 0 10 0 0 1
Sometimes write it as
x
y
z
x
y
z
x xy y
z z
p Tp
aa
a
aaa
T , ,x y za a a
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ScalingUse fixed point parameter as referenceScaling in one direction means scale in each direction elementThe scaling equation are:
For the homogeneous form
where
We may say that the inverse form of scaling is :
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Rotation2D rotation is actual 3D rotation around Z axisFor general equations:
.cos .sin
.sin .cosx x yy x yz z
q qq q
z
z
z
For the Homogeneous equation, let denotes R is a rotation matrix around z axisThen the homogeneous form for rotation is p =R pwhere
cos sin 0 0sin cos 0 0
R0 0 1 00 0 0 1
q qq q
q
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x
For rotation about x-axis the rotation matrix R
1 0 0 00 cos sin 0
R R0 sin cos 00 0 0 1
and for rotation about y-axis R
cos 0 sin 00 1 0 0
R Rsin 0 cos 00 0 0 1
x
x
y
y y
q qq
q q
q q
qq q
-1 T
For inverse rotation transformation in each axis: Rotate to the same axis with equivalent angle backward:
R = R R
We call a matrix whose inverse is equal to its transpose is call an or
q q q
thogonal matrix
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Shear
• Let pull in top right edge and bottom left edge– Neither y nor z are changed– Call x shear direction
Shear
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Computation of the shear matrix
cot xx x yy yz z
q
Leading to the shearing matrix
1 cot 0 00 1 0 0
H0 0 1 00 0 0 1
x
x x
q
q
Inverse of the shearing matrix: shear in the opposite direction
1H Hx x x xq q
x
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Shear in other direction
cot
1 0 0 0cot 1 0 0
H0 0 1 00 0 0 1
y
yy y
x xy y x
z z
q
Shear in y-axis direction when look on xy-plane
cot1 0 0 00 1 0 0
Hcot 0 1 0
0 0 0 1
z
z zz
x xy yz z x q
Shear in z-axis direction when look on zx-plane
What about Hy and Hz when look in yz plane ?
Shearing general form
• shear with reference line
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(0,0)
(0,1) (1,1)
(1,0) (0,0) (1,0)
(2,1) (3,1)
x x
y y
(0,0)
(0,1) (1,1)
(1,0) (1/2,0) (3/2,0)
(1,1) (2,1)
x x
y y
yref =-1
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Concatenation of transformation
Pipeline transformation
Application of transformations one at a time
The sequence of transformation isq=CBAp
It can be seen that one do first must be near most to inputWe may proceed in two stepsCalculate total transformation matrix:
M=CBA
Matrix operation : q=Mp
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Derive example of M: rotation about a fixed point
<- This is what we try to do.
Transformation processes
Sequence of transformation
Rotation of a cube about its center
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General rotation
Rotation of a cube about the x-axis
Rotation of a cube about the z-axis. The cube is show (a) before rotation, and (b) after rotation
Rotation of a cube about the y-axis
R R R Rx y z
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The Instance transformation• From the example object, 2 options to do with
object– Define object to its vertices and location
with the desire orientation and size– Define each object type once at a
convenience size, place and orientation , next move to its place
• Object in the scene is instance of the prototype
• If apply transformation, called instance transformation
• We can use database and identifier for eachScene of simple objects
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Instance transformation
M=TRS //instant transformation equation of object
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Rotation around arbitrary axis
• Input parameter– Object point– Vector (line segment or 2 points)– Angle of rotation
• Idea– Translate to origin first T(-P0)– Rotate
• q-axis component • q around 1 component axis, let say z
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<- Convert u to unit length vector
<- Shift to origin (T(-P)), end shift back (T(P))
<- Individual axis rotation z first
Rotation of a cube about an arbitrary axis
R = Rx(-qx).Ry(-qy).Rz(q).Ry(qy).Rx(qx)
Let rotation axis vector is u and
u = P2 – P1
Movement of the fixed point to the origin
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Problem: How we fine θx and θy ?
<- From v, unit vector
<-<-
f = angle of vector respect to origin
Sequence of rotations
Direction angles
2 2 2x y z 1a a a
cos cos cos2 2 2x y z 1f f f
coscos
cos
x x
y y
z z
f af a
f a
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<- For rotation around X, project vector to plane XZ (X=0) meanwhile projection on YZ plane with length d
->
Computation of the x rotation
Computation of the y rotation
2 2y zd a a
1 0 0 0
0 0
0 0
0 0 0 1
yz
x xy z
d dR
d d
aa
qa a
0 0
0 1 0 00 0
0 0 0 1
x
y yx
d
Rd
a
qa
Finally we calculate all the matrices to find
0 0M T p T px x y y z y y x xR R R R Rq q q q q
• Example: rotate around origin to point (1,2,3) 45 degree
Normalized vector (1, 2, 3)
Z axis rotation (0,0,1,0) 45 degree => θz = 45o
X axis (1,0,0,0) rotation angle is Y axis rotation
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Last take back to point
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OpenGL transformation matrices
• Use glMatrixMode function for selected operation to apply
• Most graphic system use Current Transformation Matrix (CTM) for any transformation
Current transformation matrix (CTM)
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CTM step
• Applied Identity matrix unless operate every round– C <- I
• Applied Translation– C <- CT
• Applied Scaling– C <- CS
• Applied Rotation– C <- CR
• Or postmultiply with M– C <- M or C <- CM
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CTM: Translation, rotation and scaling
• CTM may be viewed as the part of all primitive product GL_MODELVIEW and GL_PROJECTION
• From function of glMatrixMode
Model-view and projection matrices
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Rotation about a fixed point matrix step
glMatrixMode(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);
//rotate 45 deg. around a vector line 1,2,3// with fixed point of (4, 5, 6)
Modeling more than 1 independent object
• use paradigm of glPushMatrix() and glPopMatrix()
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The quaternion• Extension of complex number• Useful for animation and hardware
implementation of rotation• Consider the complex number
• Rotate point (a, b) around z axis with θ radian• quaternion is equivalent to rotate in 3D space
ic a ib re q
2 2 1; tan
ic a ib reWhere
br a ba
q
q
(a, b)
θ
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Quaternion form
• q0 define rotation angle• q define point
0 1 2 3 0
1 2 3 1 2 3
2 2 2
, , , ,q
q , ,
1
a q q q q q
whereq q q q i q j q k
and
i j k i j k
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Multiplication properties
• Identity– (1, 0)
• Inverse
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Rotation with quaternion• Let p=(0,p) // point in space with p= (x,y,z)• And unit quaternion r and its inverse r-1
// v is a unit vector // inverse of quaternion
r : rotation of θ around a unit vector, thus rotate of point p around v with θ and p’ is
if v= (0,0,1) //rotate around z axis, p’ will be
Quaternion to Rotation MatrixReplace:
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Using quaternion in OpenGL
• can use only in the rotation transformation• the other transformations still use the matrix
manipulation.• when use in OpenGL
– multiplay the object first and transform the result to matrix form
– use at the display state
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Case study Trackball
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Exercise4.1Consider the solution of either constant-coefficient linear differential
or difference equations (recurrences). Show that the solutions of the homogeneous equations form a vector space. Relate the solution for a particular in homogeneous equation to an affine space.
4.2Show that the following sequences commute:a. A rotation and a uniform scaling b. Two rotations about the same axis c. Two translations
4.3Write a library of functions that will allow you to do geometric programming. Your library should contain functions for manipulating the basic geometric types (points, lines, vectors) and operations on those types, including dot and cross products. It should allow you to change frames. You can also create functions to interface with OpenGL, so that you can display the results of geometric calculations.
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4.4 If we are interested in only two-dimensional graphics, we can use three-dimensional homogeneous coordinates by representing a point as p = [x y 1]T and a vector as v =[a b 0]T. Find the 3x3 rotation, translation, scaling, and shear matrices. How many degrees of freedom are there in an affine transformation for transforming two-dimensional points?
4.5 We can specify an affine transformation by considering the location of a small number of points both before and after these points have been transformed. In three dimensions, how many points must we consider to specify the transformation uniquely? How does the required number of points change when we work in two dimensions?
4.6 How must we change the rotation matrices if we are working in a left-handed system and we retain our definition of a positive rotation?
4.7 Show that any sequence of rotations and translations can be replaced by a single rotation about the origin, followed by a translation.
4.8 Derive the shear transformation from the rotation, translation, and scaling transformations.
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4.9 In two dimensions, we can specify a line by the equation y = mx + b. Find an affine transformation to reflect two- dimensional points about this line. Extend your result to reflection about a plane in three dimensions,
4.10 In Section 4.8 we showed that an arbitrary rotation matrix could be composed from successive rotations about the three axes. How many ways can we compose a given rotation if we can do only three simple rotations? Are all three of the simple rotation matrices necessary?
4.11 Add shear to the instance transformation. Show how to use this expanded instance transformation to generate parallelepipeds from a unit cube.
4.12 Find a homogeneous-coordinate representation of a plane.4.13 Determine the rotation matrix formed by glRotate. That is, assume that
the fixed point is the origin and that the parameters are those of the function.
4.14 Write a program to generate a Sierpinski gasket as follows. Start with a white triangle. At each step, use transformations to generate three similar triangles that are drawn over the original triangle, leaving the center of the triangle white and the three corners black.
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4.15 Start with a cube centered at the origin and aligned with the coordinate axes. Find a rotation matrix that will orient the cube symmetrically, as shown in Figure 4.65.
4.16 We have used vertices in three dimensions to define objects such as three-dimensional polygons. Given a set of vertices, find a test to determine whether the polygon that they determine is planar.
4.17 Three vertices determine a triangle if they do not lie in the same line. Devise a rest for co-linearity of three vertices.
4.18 We defined an instance transformation as the product of a translation, a rotation, and a scaling. Can we accomplish the same effect by applying these three types of transformations in a different order?
4.19 Write a program that allows you to orient the cube with one mouse button, to translate it with a second, and to zoom in and out with a third.
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4.20 Given two nonparallel three-dimensional vectors u and r, how can we form an orthogonal coordinate system in which u is one of the basis vectors?
4.21 An incremental rotation about the z-axis is determined by the matrix
What negative aspects are there if we use this matrix for a large number of steps? Hint: consider points a distance of 1 from the origin. Can you suggest a remedy?
4.22 Find the quaternion for 90-degree rotations about the x- and y-axes. Determine their product.
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4.23 Determine the rotation matrix
find the corresponding quaternion.4.24 Redo the trackball program using quaternion instead of
rotation matrices.4.25 Implement a simple package in C++ that can do the required
mathematical operations for transformations. Such a package might include matrices, vectors, and frames