Geometric Objects and Transformations Geometric Entities Representation vs. Reference System Geometric 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 segment Directed 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 - - -

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.

Line in an Affine Space

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

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.

Dot Product and Projection u

Cross Product  Right-handed coordinate system

Formation of a Plane  Normal vector:

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 w

Coordinate Systems

Changes of Coordinate Systems

Changes of Coordinate Systems (2)

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

Homogeneous Coordinates  Represent frame change (translation)  Avoid point vs. vector confusion

Example of Change in Frames

Example (continued)

Camera and World Frames In OpenGL, the model-view matrix positions the world frame relative to the camera frame.

One Frame of Cube Animation Tasks: 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) Outward facing

Vertex-List Representation of a Cube

Bilinear Interpolation

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 X’, y’

Application of Pipeline Transformation

Rotation of a Cube About Its Center

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

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

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 centered at the origin and aligned with the coordinate axes. Find a rotation matrix that will orient the cube symmetrically, as shown here.