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Chapter 4. Geometric Objects and Transformation. Lines. Consider all points of the form P( a )=P 0 + a d Set of all points that pass through P 0 in the direction of the vector d. Parametric Form. This form is known as the parametric form of the line

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Chapter 4

Chapter 4

Geometric Objects and Transformation



  • Consider all points of the form

    • P(a)=P0 + ad

    • Set of all points that pass through P0 in the direction of the vector d

Parametric form

Parametric Form

  • This form is known as the parametric form of the line

    • More robust and general than other forms

    • Extends to curves and surfaces

  • Two-dimensional forms

    • Explicit: y = mx +h

    • Implicit: ax + by +c =0

    • Parametric:

      x(a) = ax0 + (1-a)x1

      y(a) = ay0 + (1-a)y1

Rays and line segments

Rays and Line Segments

If a >= 0, then P(a) is the ray leaving P0 in the direction d

If we use two points to define v, then

P( a) = Q + a (R-Q)=Q+av

=aR + (1-a)Q

For 0<=a<=1 we get all the

points on the line segment

joining R and Q

Dot and cross products

Dot and Cross: Products

Three dimensional primitives

Three-Dimensional Primitives

3D curves

3D surfaces

Volumetric Objects

Hollow objects

Objects can be specified by vertices

Simple and flat polygons (triangles)

Constructive Solid Geometry (CSG)

Constructive solid geometry

Constructive Solid Geometry



  • Until now we have been able to work with geometric entities without using any frame of reference, such a coordinate system

  • Need a frame of reference to relate points and objects to our physical world.

    • For example, where is a point? Can’t answer without a reference system

    • World coordinates

    • Camera coordinates

Confusing points and vectors

Confusing Points and Vectors




can place anywhere


Consider the point and the vector

P = P0 + b1v1+ b2v2+….+bnvn

v=a1v1+ a2v2+….+anvn

They appear to have the similar representations

P=[b1 b2 b3] v=[a1 a2 a3]

which confuse the point with the vector

A vector has no position

A single representation

A Single Representation

If we define 0•P = 0 and 1•P =P then we can write

v=a1v1+ a2v2+a3v3 = [a1 a2a30][v1 v2 v3 P0] T

P = P0 + b1v1+ b2v2+b3v3= [b1 b2b31][v1 v2 v3 P0] T

Thus we obtain the four-dimensional homogeneous coordinate representation

v = [a1 a2a30] T

P = [b1 b2b31] T

Homogeneous coordinates

Homogeneous Coordinates

The general form of four dimensional homogeneous coordinates is

p=[x y x w] T

We return to a three dimensional point (for w0) by




If w=0, the representation is that of a vector

Note that homogeneous coordinates replaces points in three dimensions by lines through the origin in four dimensions

Homogeneous coordinates and computer graphics

Homogeneous Coordinates and Computer Graphics

  • Homogeneous coordinates are key to all computer graphics systems

    • All standard transformations (rotation, translation, scaling) can be implemented by matrix multiplications with 4 x 4 matrices

    • Hardware pipeline works with 4 dimensional representations

    • For orthographic viewing, we can maintain w=0 for vectors and w=1 for points

    • For perspective we need a perspective division

Representing a mesh

Representing a Mesh





















  • Consider a mesh

  • There are 8 nodes and 12 edges

    • 5 interior polygons

    • 6 interior (shared) edges

  • Each vertex has a location vi = (xi yi zi)

Inward and outward facing polygons

Inward and Outward Facing Polygons

The order {v0, v3, v2, v1} and {v1, v0, v3, v2} are equivalent in that the same polygon will be rendered by OpenGL but the order {{v0, v1, v2, v3} is different

The first two describe outwardly

facing polygons

Use the right-hand rule =

counter-clockwise encirclement

of outward-pointing normal

OpenGL treats inward and

outward facing polygons differently

Geometry versus topology

Geometry versus Topology

  • Generally it is a good idea to look for data structures that separate the geometry from the topology

    • Geometry: locations of the vertices

    • Topology: organization of the vertices and edges

    • Example: a polygon is an ordered list of vertices with an edge connecting successive pairs of vertices and the last to the first

    • Topology holds even if geometry changes

Bilinear interpolation

Bilinear Interpolation

Assuming a linear variation, then we can make use of the

same interpolation coefficients in coordinates for the

interpolation of other attributes.

Scan line interpolation

Scan-line Interpolation

A polygon is filled only when it is displayed

It is filled scan line by scan line

Can be used for other associated attributes with each vertex

General transformations

General Transformations

A transformation maps points to other points and/or vectors to other vectors

Linear function transformation

Linear Function (Transformation)

Transformation matrix for homogeneous coordinate system:

Affine transformations 1 2

Affine Transformations – 1/2

  • Line preserving

  • Characteristic of many physically important transformations

    • Rigid body transformations: rotation, translation

    • Scaling, shear

  • Importance in graphics is that we need only transform endpoints of line segments and let implementation draw line segment between the transformed endpoints

Affine transformations 2 2

Affine Transformations – 2/2

Every linear transformation (if the corresponding matrix is nonsingular) is equivalent to a change in frames

However, an affine transformation has only 12 degrees of freedom because 4 of the elements in the matrix are fixed and are a subset of all possible 4 x 4 linear transformations





  • Move (translate, displace) a point to a new location

  • Displacement determined by a vector d

    • Three degrees of freedom

    • P’=P+d

Rotation 2d 1 2

Rotation (2D) – 1/2

x´ = r cos (f + q) = rcosf cosq - rsinf sinq

y´ = r sin (f + q) = rcosf sinq + rsinf cosq

x´ =x cos q –y sin q

y´ = x sin q + y cos q

x = r cos f

y = r sin f

  • Consider rotation about the origin by q degrees

    • radius stays the same, angle increases by q

Rotation 2d 2 2

Rotation (2D) – 2/2

Using the matrix form:

There is a fixed point

Could be extended to 3D

Positive direction of rotation is counterclockwise

2D rotation is equivalent to 3D rotation about the z-axis

Non rigid body transformation

(Non-)Rigid-Body Transformation


Translation and rotation are rigid-body transformation






Uniform and non-uniform


Expand or contract along each axis (fixed point of origin)



sx = -1 sy = 1


sx = -1 sy = -1

sx = 1 sy = -1

corresponds to negative scale factors

Transformation in homogeneous coordinates

Transformation in Homogeneous Coordinates

With a frame, each affine transformation is represented by a 44 matrix of the form:



note that this expression is in

four dimensions and expresses

that point = vector + point

Using the homogeneous coordinate representation in some frame

p=[ x y z 1]T

p´=[x´ y´ z´ 1]T

d=[dx dy dz 0]T

Hence p´= p + dor




Translation matrix

Translation Matrix

We can also express translation using a

4 x 4 matrix T in homogeneous coordinates

p´=Tp where

This form is better for implementation because all affine transformations can be expressed this way and multiple transformations can be concatenated together

Rotation about the z axis

Rotation about the Z axis

x´=x cos q –y sin q

y´= x sin q + y cos q


  • Rotation about z axis in three dimensions leaves all points with the same z

    • Equivalent to rotation in two dimensions in planes of constant z

    • or in homogeneous coordinates


Rotation matrix

Rotation Matrix

Rotation about x and y axes

Rotation about X and Y axes

  • Same argument as for rotation about z axis

    • For rotation about x axis, x is unchanged

    • For rotation about y axis, y is unchanged

Scaling matrix

Scaling Matrix







  • Although we could compute inverse matrices by general formulas, we can use simple geometric observations

    • Translation: T-1(dx, dy, dz) = T(-dx, -dy, -dz)

    • Rotation: R-1(q) = R(-q)

      • Holds for any rotation matrix

      • Note that since cos(-q) = cos(q) and sin(-q)=-sin(q)

        R-1(q) = R T(q)

    • Scaling: S-1(sx, sy, sz) = S(1/sx, 1/sy, 1/sz)



We can form arbitrary affine transformation matrices by multiplying together rotation, translation, and scaling matrices

Because the same transformation is applied to many vertices, the cost of forming a matrix M=ABCD is not significant compared to the cost of computing Mp for many vertices p

The difficult part is how to form a desired transformation from the specifications in the application

Order of transformations

Order of Transformations

Note that matrix on the right is the first applied

Mathematically, the following are equivalent

p´ = ABCp = A(B(Cp))

Note many references use column matrices to present points. In terms of column matrices


Rotation about a fixed point and about the z axis

Rotation about a Fixed Point and about the Z axis


General rotation about the origin

General Rotation about the Origin

R(q) = Rx() Ry() Rz()


, ,are called the Euler angles



Note that rotations do not commute

We can use rotations in another order but

with different angles



A rotation by q about an arbitrary axis can be decomposed into the concatenation of rotations about the x, y, and z axes

Rotation about an arbitrary axis 1 2

Rotation about an Arbitrary Axis – 1/2

Final rotation matrix:

Normalize u:

Rotation about an arbitrary axis 2 2

Rotation about an Arbitrary Axis – 2/2

Rotate the line segment to the plane of y=0, and the line segment is foreshortened to:

Rotate clockwise about the y-axis, so:

Final transformation matrix:

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