DREAM IDEA PLAN IMPLEMENTATION 1 2

DREAM IDEA PLAN IMPLEMENTATION 1

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Introduction to Computer Graphics Present to: Amirkabir University of Technology (Tehran Polytechnic) & Semnan University Dr. Kourosh Kiani Email: kkiani 2004@yahoo. com Email: Kourosh. kiani@aut. ac. com Web: aut. ac. com 3

Amirkabir & Semnan University Computer & Electrical Faculty Lecture 07 3 D viewing 4

General 3 D Viewing Pipeline • • • Modeling coordinates (MC) World coordinates (WC) Viewing coordinates (VC) Projection coordinates (PC) Normalized coordinates (NC) Device coordinates (DC) 5

Viewing Pipeline 6

Viewing Pipeline 7

Object Coordinates 8

World Coordinates 9

Object Coordinates 10 World Coordinates

More Complex Models 11

Projections We will look at several planar geometric 3 D to 2 D Projection: • Parallel Projections Orthographic Oblique • Perspective Projection of a 3 D object is defined by straight projection rays (projectors) emanating from the Center Of Projection (COP) passing through each point of the object and intersecting the projection plane. 12

Taxonomy of Geometric Projections geometric projections parallel orthographic perspective axonometric Top trimetric Side isometric cavalier cabinet dimetric Front oblique 13 single-point two-point three-point

Parallel Projections 14

Parallel Projections 15

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Orthographic projection. xp = x yp = y zp = 0 wp = 1 In practice, we can let M = I and set the z term to zero later 19

Axonometric Transformation • Orthogonal projection that displays more than one face of an object – Projection plane is not normal to a principal axis, but DOP is perpendicular to the projection plane – Isometric, dimetric, trimetric – How many angles of a a cube’s corner are equal? • None: trimetric • Two: dimetric • Three: Isometric 20

Axonometric orthographic projections use planes of projection that are not normal to a principal axix ( they therefore show multiple face of an object. ) Isometric projection: projection plane normal makes equal angles with each principal axis. DOP vector: [1 1 1]. All 3 axis are equally foreshortened allowing measurements along the axes to be made with the same scale. 21

Axonometric Transformation 22

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Oblique Projections Projection plane normal and the Direction Of Projection (DOP) differ. Plane of projection is normal to a Principle axis Projectors are not normal to the projection plane 24

Oblique Projections 25

Orthographic Projections Oblique Projections 26

Oblique Projections 27

Oblique Projections 28

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Oblique Projections When : Cavelier Lines perpendicular to the projection plane are not foreshortend. When : Cabinet Lines perpendicular to the projection plane are foreshortened by half. is typically 30 or 45 30

Oblique Projections: Cabinet 31

Oblique Projections: Cavalier 32

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• Cavalier projection : – Preserves lengths of lines perpendicular to the viewing plane – 3 D nature can be captured but shape seems distorted – Can display a combination of front, side, and top views • Cabinet projection: – Lines perpendicular to the viewing plane project at 1/2 of their length – A more realistic view than the Cavalier projection – Can display a combination of front, side, and top views 34

View plane 35

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View plane 37

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Perspective Projections 39

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Perspective Projections Distance from COP to projection plane is finite. The projectors are not parallel and we specify a Center of Projection (COP). Center of Projection is also called the Perspective Reference Point COP = PRP 41

Perspective Projections 42

Perspective Projections 43

Perspective Projections 44

Frustum view volume yv Centre of View volume projection for perspective projection View window zv Back plane 45 Front plane xv

If the distance from the center of projection to the projection plane is finite, the projection is perspective. If the distance is infinite, the projection is parallel. A Projectors 46 A B B Center of projection A Center of projection at infinity

Orthographic Projections 47 COP Perspective Projections

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Vanishing points image plane vanishing point camera center • Vanishing point – projection of a point at infinity 49

Vanishing points (2 D) image plane vanishing point camera center line on ground plane 50

Vanishing points image plane vanishing point V camera center C line on ground plane • Properties – Any two parallel lines have the same vanishing point v – The ray from C through v is parallel to the lines – An image may have more than one vanishing point • in fact every pixel is a potential vanishing point 51

Vanishing Point Vanishing Line Vanishing Point Projection Center COP View Plane 52

View Plane Projection Center COP es are parallel to projection plane, there is no vanishing point. 53

Vanishing point • Parallel lines that are not parallel to the viewing plane, converge to a vanishing point • A vanishing point is the projection of a point at infinite distance 54

1. One-point perspective 2. Two-point perspective 3. Three-point perspective 55

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Three-point perspective 60 Two-point perspective One-point perspective

One-point perspective 61

Two-point perspective projection 62

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Vector Specification • Most flexible method of viewing specification • View plane may be anywhere with respect to the world • Locate the View Plane (I. e. Projection Plane) by: – View reference point (VRP) – View-plane normal (VPN) (n axis) 65

VRC Coordinate System • Viewing Reference Coordinate (VRC) System – n-axis - along View plane normal (VPN) – v-axis – projection of View Up Vector (VUP) – u-axis – Right hand coordinate system 66

Viewing Window • Min/Max u and v ranges • Need not be symmetrical around VRP 67

Center of Projection • Projection-Reference Point (PRP=COP) 68

Direction of Projection DOP • Projection-Reference Point (PRP) The VRC is a right-handed system made up of u, v, n. CW is the center of the window. Infinite parallelepiped view volume of parallel orthographic projection. 69

Oblique Parallel Projection • Vector from PRP to Center of the Window not parallel to VPN 70

Perspective View Volumes 71 Truncated view volume for perspective projection

Orthogonal View Volumes Truncated view volume for an orthographic parallel projection 72

Oblique Projections Truncated view volume for an oblique parallel projection 73

Examples of 3 D viewing: Parallel Projections Viewing situation that creates the previous figure. The PRP could be any point with x=8 and y= 8. 74

Examples of 3 D viewing: Parallel Projections Front parallel projection Front view VRP(WC) (0, 0, 0) VPN(WC) (0, 0, 1) VUP(WC) (0, 1, 0) PRP(VRC) (8, 8, 100) window(VRC) (-1, 17, -1, 17) projection type : parallel

Examples of 3 D viewing: Parallel Projections Side view VRP(WC) (0, 0, 54) VPN(WC) (1, 0, 0) VUP(WC) (0, 1, 0) PRP(VRC) (12, 8, 16) window(VRC) (-1, 25, -5, 21) projection type: parallel The viewing situation. 76

Examples of 3 D viewing: Parallel Projections Parallel projection from the side the house 77 Side view VRP(WC) (0, 0, 54) VPN(WC) (1, 0, 0) VUP(WC) (0, 1, 0) PRP(VRC) (12, 8, 16) window(VRC) (-1, 25, -5, 21) projection type: parallel

Examples of 3 D viewing: Perspective Projections VRP(WC) (0, 0, 54) VPN(WC) (0, 0, 1) VUP(WC) (0, 1, 0) PRP(VRC) (8, 6, 30) window(VRC) (-1, 17, -1, 17) projection type: perspective Centered perspective projection of a house. 78

Examples of 3 D viewing: Perspective Projections The viewing situation of the previous slide. 79

X or Y P(X, Y, Z) PP Xp or Yp O (COP) d 80 Z

X or Y P(X, Y, Z) PP Xp or Yp o COP d 81 Z

X or Y PP COP L P’(X, Y, Z) Q P(X, Y, Z) Z O (0, 0 Zp) Parametric equation of the line L between COP and P: 82

Let the direction vector from Q be the distance from to COP be to COP. and Then The coordinates of any point on line L is: Using the condition PP: 83 at the intersection of line L and plane

Generalized formula of perspective projection matrix: 84

X or Y PP COP L P’(X, Y, Z) Q P(X, Y, Z) Z (0, 0 Zp) O Special cases from the generalized formulation of the perspective projection matrix Matrix type Morth Mper M’per 85 Zp 0 d 0 Q infinity d d [ d x dy [0 0 dz ] -1 ] If Q is finite, Mgen defines a one-point perspective in the above cases

The Viewing Volume • Volume defined by: – Near Plane (n) – Far Plane (f) – Width (W) – Height (H) • For Orthographic Projection 86

The Viewing Volume 87

The Viewing Volume 88

The Viewing Volume • Volume defined by: – Near Plane (n) – Far Plane (f) – Fields of view (fov) • For Orthographic Projection 89

The Viewing Volume 90

View Volume Center of Projection View Plane 91

The Viewing Volume 92

Questions? Discussion? Suggestions ?

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