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Geometric Objects and Transformations

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  1. Geometric Objects and Transformations Geometric Entities Representation vs. Reference System Geometric ADT (Abstract Data Types)

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

  3. 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

  4. Geometric View • Point: a location in space • Scalar: real number • Vector: directed line segment Directed Line Segment That Connects Points

  5. Identical Vectors

  6. Combination of Directed Line Segments

  7. A Dangerous Representation of a Vector

  8. 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

  9. Point-Point Subtraction

  10. 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;

  11. - - - Use of the Head-to-Tail Axiom

  12. 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.

  13. Line in an Affine Space

  14. 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.

  15. Affine Addition

  16. 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

  17. 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.

  18. u Dot Product and Projection

  19. Cross Product • Right-handed coordinate system

  20. Formation of a Plane • Normal vector:

  21. 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.

  22. Primitives in Three Dimensions

  23. w Vector Derived From Three Basis Vectors

  24. Coordinate Systems

  25. Changes of Coordinate Systems

  26. Changes of Coordinate Systems (2)

  27. 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.

  28. Translation of a Basis • However, a simple translation of the origin, or change of frame cannot be represented in this way. • Homogeneous coordinates.

  29. Example of Change of Representation

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

  31. Example of Change in Frames

  32. Example (continued)

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

  34. One Frame of Cube Animation • Tasks: • Modeling (8 vertices, 12 edges) • Converting to the camera frame • Clipping • Projecting • Removing hidden surfaces • Rasterizing

  35. 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

  36. Vertex-List Representation of a Cube

  37. Bilinear Interpolation

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

  39. Scan-Line Interpolation OpenGL Approach

  40. Transformation

  41. Translation

  42. Two-Dimensional Rotation

  43. Rotation About a Fixed Point

  44. Three-Dimensional Rotation

  45. Non-Rigid Body Transformation

  46. Uniform and Non-uniform Scaling

  47. Effect of Scale Factor

  48. Reflection

  49. Shear

  50. X’, y’ Computation of the Shear Matrix