Geometric Objects and Transformations

Geometric EntitiesRepresentation vs. Reference SystemGeometric ADT (Abstract Data Types)

Goals Coordinate-free representation of objects Homogeneous coordinates Distinction of an object from its representation

Scalars, Points, and Vectors Geometric View Mathematical View

– Vector– Affine Spaces

Computer-Science View Other issues

– Geometric ADTs– Lines– Affine Sums– Convexity– Dot and Cross Products – Planes

Geometric View Point: a location in space Scalar: real number Vector: directed line segmentDirected Line Segment That Connects Points

Identical Vectors

Combination of Directed Line Segments

A Dangerous Representation of a Vector

Mathematical View Vector space: vectors and scalars Operations

– Scalar-vector multiplication– Vector-vector addition

Affine space: vector space + point– Vector-point addition (or point-point subtraction)

Euclidean space: affine space + measure of size or distance

Point-Point Subtraction

Computer-Science View Scalars, points, and vectors as abstract data types (ADTs) ADT

– Object-oriented– Data and operations

Object declarations

vector u,v;point p,q;scalar a,b;

q = p + a * v;

Use of the Head-to-Tail Axiom

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Lines The sum of a point and a vector (or subtraction of

two points) leads to the notion of a line in an affine space.

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Affine Operations In an affine space, scalar-vector multiplication

vector-vector addition are defined. However, point-point addition and scalar-point multiplication are not.

Affine addition has certain elements of the above two latter operations.

Affine Addition

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Convexity A convex object is one for which any point lying

on the line segment connecting any two points in the object is also in the object.

Line segment that connects two points

Convex Hull

Geometrically, a convex hull is the set of points that we form by stretching a tight fitting surface over the given set of points – shrink wrapping the points.

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Three-Dimensional Primitives Three features characterize three-dimensional objects

that fit well with existing graphics hardware and software:– Objects described by surfaces and hollow – 2D primitives

modeling 3D primitives;– Objects can be specified through vertices – efficient

implementation;– The objects either are composed of or can be approximated by

flat convex polygons.

Primitives in Three Dimensions

Vector Derived From Three Basis Vectors

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Changes of Coordinate Systems

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Rotation and Scaling of a Basis The changes in basis leave the origin unchanged.

We can use them to represent rotation and scaling of a set of basis vectors to derive another basis set.

Translation of a Basis However, a simple translation of the origin, or

change of frame cannot be represented in this way.

Homogeneous coordinates.

Example of Change of Representation

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Camera and World FramesIn OpenGL, the model-view matrix positions the worldframe relative to the camera frame.

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One Frame of Cube AnimationTasks:

• Modeling (8 vertices, 12 edges) • Converting to the camera frame • Clipping • Projecting • Removing hidden surfaces • Rasterizing

Traversal of the Edges of a Polygon

Counterclockwise:(0,3,2,1)(3,2,1,0)(2,1,0,3)(1,0,3,2)

Outwardfacing

Vertex-List Representation of a Cube

Bilinear Interpolation

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Projection of Polygon

A polygon is first projected onto the two-dimensional plane and then filled with colors.

Scan-Line Interpolation

OpenGL Approach

Transformation

Translation

Two-Dimensional Rotation

Rotation About a Fixed Point

Three-Dimensional Rotation

Non-Rigid Body Transformation

Uniform and Non-uniform Scaling

Effect of Scale Factor

Reflection

Shear

Computation of the Shear Matrix

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Application of Pipeline Transformation

Rotation of a Cube About Its Center

Rotation of a Cube About Its Center – Sequence of Transformation –

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Rotation of a Cube About the z Axis

Rotation of a Cube About y Axis

Rotation of a Cube About The x Axis

Scene of Simple Objects

Instance transformation

Instead of defining each object and its size, position, orientation, we define the object types and create each instance with its size, position, orientation.

Instance Transformation

Rotation of a Cube About an Arbitrary Axis

Movement of The Fixed Point to Origin

Sequence of Rotations

Direction Angles

axes. theand

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zz

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xx

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cos

cos

Computation of the x Rotation

Computation of the y Rotation

Current Transformation Matrix (CTM)

Model-View and Projection Matrices

CTM: current transformation matrix

Symmetric Orientation of Cube Start with a cube centeredat the origin and alignedwith the coordinate axes.Find a rotation matrix thatwill orient the cube symmetrically, as shown here.