The Geometry of Virtual Worlds.pdf
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The document discusses geometric models in virtual worlds. It covers topics like representing objects as meshes of triangles, transforming objects by changing their position through translation and changing their orientation through rotation. Rotation in 3D is more complex than translation and can be represented using rotation matrices. The document provides examples of how 2D rotation matrices work and the constraints they must satisfy to represent a pure rotation without other transformations like scaling, shearing or mirroring.
The Geometry of Virtual Worlds.pdf
- 1. The Geometry of Virtual Worlds Mrs. Rashmi Bhat Assistant Professor St. John College of Engineering and Management, Palghar
- 2. Topics to discuss Geometric Models Changing Position and Orientation Axis-Angle Representations of Rotation Viewing Transformations Chaining the transformations 2 Prepared by Mrs. Rashmi Bhat
- 3. Geometric Models • A model can be any object in a Virtual World. • e.g. Chair, a tree, wall of room, user's avatar etc. • Objects can be transformed • Change in position • Change in orientation Prepared by Mrs. Rashmi Bhat 3 Change in position Change in Orientation Object’s first View Object’s Top View Object’s another View
- 4. Geometric Models • A virtual world can be represented in 3D Euclidean space with cartesian coordinates • Let represents a virtual world • Every point is represented as a triple of real-valued coordinates: (𝒙, 𝒚, 𝒛) • We follow right handed coordinate system Prepared by Mrs. Rashmi Bhat 4
- 5. Geometric Models Prepared by Mrs. Rashmi Bhat 5 Fig. Coordinate System
- 6. Geometric Models Prepared by Mrs. Rashmi Bhat 6 Fig. Coordinate Axes of Virtual World
- 7. Geometric Models • Geometric Models • Are made of surfaces or solid regions in • Contain infinite no. of points • Are defined in terms of primitives • Each primitive represents an infinite set of points • e.g. 3D triangle • Planar surface/ face of the triangle is specified by the coordinates of the triangle vertices ( 𝑥 , 𝑦 , 𝑧 , 𝑥 , 𝑦 , 𝑧 , (𝑥 , 𝑦 , 𝑧 )) Prepared by Mrs. Rashmi Bhat 7
- 8. Geometric Models • Geometric Models • A complicated object or body in the virtual world modelled using numerous triangles can be arranged into a mesh. Prepared by Mrs. Rashmi Bhat 8
- 9. Geometric Models • Models representation in 3D mesh-triangle provokes many questions 1. How do we specify how each triangle "looks" whenever viewed by a user in VR? 2. How do we make the object "move"? 3. If the object surface is sharply curved, then should we use curved primitives? 4. Is the interior of the object part of the model, or does the object consist only of its surface? 5. Is there an efficient algorithm for determining which triangles are adjacent to a given triangle along the surface? 6. Should we avoid duplicating vertex coordinates that are common to many neighboring triangles? Prepared by Mrs. Rashmi Bhat 9
- 10. Geometric Models • Data Structures • Choice of data structure for geometric models should depend on which operation will be performed on the model. • Most commonly used data structure is Doubly Connected Edge List(DCEL) (also known as Half-Edge data structure) • Three kinds of data elements • Face represents two dimensional part of the model • Edge represents one dimensional part of the model • Vertex represents zero dimensional part of the model Prepared by Mrs. Rashmi Bhat 10
- 11. Geometric Models Prepared by Mrs. Rashmi Bhat 11 Face Vertex Edge Fig. Part of a doubly connected edge list • Data Structures • The data structure contains pointers between adjacent faces, edges, and vertices so that algorithms can quickly traverse the model components.
- 12. Geometric Models • Inside vs Outside • The model forms a complete barrier between the inside and outside of the object. • Coherent Model • Use of high-end design tools like Blender/ Maya • Why triangles? • The simplest for algorithms to handle • Complicated primitives may lead to smaller model sizes but increases the cost of computation Prepared by Mrs. Rashmi Bhat 12
- 13. Geometric Models • Stationary vs Movable Models in Virtual World • Stationary models, which keep the same co-ordinates forever. • Movable models, which can be transformed into various positions and orientations. • Motion can be caused in following ways • Using tracking system • With the use of controllers • Self-movement according to the laws of physics Prepared by Mrs. Rashmi Bhat 13
- 14. Geometric Models • Choosing Coordinate Axes • Important to choose coordinates for the models • Put origin (0,0,0) in a convenient location • 𝑦 = 0 corresponds to the floor or sea level • 𝑥 = 0 and 𝑧 = 0 can be in the center of virtual world • For movable models the location of the origin and axes direction is very important Prepared by Mrs. Rashmi Bhat 14
- 15. Geometric Models • Viewing the models • How the model will look on VR display? • Two parts of the problem • Determine where the points in a virtual world should appear on display (Transformation) • Determining how each part of the model should appear after taking into account lighting sources and surface properties that are defined in the virtual world. (Rendering) Prepared by Mrs. Rashmi Bhat 15
- 16. Changing Position and Orientation Prepared by Mrs. Rashmi Bhat 16
- 17. Z X Y Changing Position and Orientation • Assume model is defined as a mesh of triangles. • Changing position: Translation • Changing orientation: Rotations Prepared by Mrs. Rashmi Bhat 17 (0, 0, 0) X Y Z
- 18. Translation • Consider a 3D triangle • Amount to translate the position of a given triangle be , and along the and axis respectively. • The operation of changing position is called translation. • It is given by 𝑥 , 𝑦 , 𝑧 ⟼ 𝑥 + 𝑥 , 𝑦 + 𝑦 , 𝑧 + 𝑧 𝑥 , 𝑦 , 𝑧 ⟼ 𝑥 + 𝑥 , 𝑦 + 𝑦 , 𝑧 + 𝑧 𝑥 , 𝑦 , 𝑧 ⟼ 𝑥 + 𝑥 , 𝑦 + 𝑦 , 𝑧 + 𝑧 Prepared by Mrs. Rashmi Bhat 18
- 19. Translation • Here, , denotes that becomes replaced by after the transformation. • This transformation will be applied to every triangle in a model. • If the triangles are arranged in a mesh, then apply transformation to vertices only. Prepared by Mrs. Rashmi Bhat 19
- 20. Relativity • Every transformation has two possible interpretations • Translating a triangle • Moving the origin Prepared by Mrs. Rashmi Bhat 20 𝒙 𝒚 𝒛 (𝟐𝟎, 𝟏𝟎, 𝟎) (𝟏𝟎, 𝟏𝟎, 𝟎) (𝟏𝟓, 𝟏𝟓, 𝟎) (𝟏𝟏, 𝟐, 𝟎) (𝟏, 𝟐, 𝟎) (𝟔, 𝟕, 𝟎) 𝒙 𝒚 𝒛 (𝟏𝟏, 𝟐, 𝟎) (𝟏, 𝟐, 𝟎) (𝟔, 𝟕, 𝟎) 𝒙 𝒚 𝒛 𝒙 𝒚 𝒛 (a) Original Object (b) Object moves (c) Origin moves Object moves Origin moves 𝒙𝒕, 𝒚𝒕, 𝒛𝒕 = (−𝟗, −𝟖, , 𝟎)
- 21. Relativity • Relativity • Did the object move, or did the whole world move around it? • If the origin moves, then users would move with it. • If users perceive themselves as having moved, then VR sickness might increase, even though it was the object that moved. • User’s brains make their best guess as to which type of motion occurred, and sometimes get it wrong Prepared by Mrs. Rashmi Bhat 21
- 22. Rotation • Many times we need to change the model’s orientation in the virtual world. • The operation that changes the orientation is called rotation. • Rotations in three dimensions are much more complicated than translations. Prepared by Mrs. Rashmi Bhat 22
- 23. Rotation • 2D Linear Transformations • Points in 2D virtual world have coordinates (𝑥, 𝑦) • A generic 2 × 2 matrix 𝑀 = 𝑚 𝑚 𝑚 𝑚 • Multiplication of 𝑀 with a point (𝑥, 𝑦) results in a transformed point (𝑥 , 𝑦 ) 𝑚 𝑚 𝑚 𝑚 𝑥 𝑦 = 𝑥 𝑦 the matrix multiplication yields 𝑥 = 𝑚 𝑥 + 𝑚 𝑦 𝑦 = 𝑚 𝑥 + 𝑚 𝑦 • 𝑀 is a transformation for which (𝑥, 𝑦) ⟼ (𝑥 , 𝑦 ) Prepared by Mrs. Rashmi Bhat 23 ………(Eq. 1)
- 24. Rotation • 2D Linear Transformations • Applying the 2D matrix to points • Lets take two points (1,0) and (0,1) lying one unit of distance from the origin (0,0) on 𝑥 and 𝑦 axes respectively. • Substitute these points in equation A, we get 𝑚 𝑚 𝑚 𝑚 1 0 = 𝑚 𝑚 and 𝑚 𝑚 𝑚 𝑚 0 1 = 𝑚 𝑚 If is applied to transform a model, each column of indicates precisely how each coordinate axis is changed. Prepared by Mrs. Rashmi Bhat 24 These special points select the column vectors on 𝑴
- 25. Rotation • 2D Linear Transformations • Applying the 2D matrix to points Prepared by Mrs. Rashmi Bhat 25 𝑀 = (0,1) (1,0) 𝑀 = (0,1) (−1,0) Causes a flip
- 26. Rotation • 2D Linear Transformations • Applying the 2D matrix to points Prepared by Mrs. Rashmi Bhat 26 𝑀 = (0,2) (2,0) 𝑀 = (0,2) (1,0) Causes scaling Causes aspect ratio distortion
- 27. Rotation • 2D Linear Transformations • Applying the 2D matrix to points Prepared by Mrs. Rashmi Bhat 27 𝑀 = (0, −1) (−1,0) 𝑀 = (1,1) (1,1) Causes a mirror image 2D shape collapses into 1D
- 28. Rotation • 2D Linear Transformations • Applying the 2D matrix to points Prepared by Mrs. Rashmi Bhat 28 𝑀 = (1,1) (1,0) 𝑀 = (0,1) (1,1) Causes a skew in X direction Causes a skew in Y direction
- 29. Rotation • Which matrices produce rotations? • Matrix must satisfy the following rules to produce rotation: 1. No stretching of axes 2. No shearing 3. No mirror images Prepared by Mrs. Rashmi Bhat 29
- 30. Rotation Rule 1: No stretching of axes • To satisfy 1st rule, the columns of must have unit length and • Scaling and shearing transformation violates this rule Rule 2: No Shearing • To satisfy 2nd rule, the coordinate axes must remain perpendicular. • The columns of indicates how axes are transformed, this rule implies that their inner (dot) product is zero. Prepared by Mrs. Rashmi Bhat 30 ………(Eq. 2) ………(Eq. 3)
- 31. Rotation Rule 3: No Mirror Images • To satisfy 3rd rule, the determinant of is positive • After satisfying first two rules, only possible remaining determinants are and . • Rule implies that • Mirror image results in det 𝑀 = −1 Prepared by Mrs. Rashmi Bhat 31 ………(Eq. 4)
- 32. Rotation • 1st constraint indicate that each column of must be chosen so that, its components lie on a unit circle, centered at origin. and • Instead of and , use the notation of matrix components. Prepared by Mrs. Rashmi Bhat 32 (0,0) 1
- 33. Rotation • Let and , substitute this into from Eq. 1 and by applying Eq. 3 and Eq. 4, we get and • Hence, Eq. 1 becomes • The full range of allowable rotations can be generated by allowing to range from to . Prepared by Mrs. Rashmi Bhat 33
- 34. Rotation Prepared by Mrs. Rashmi Bhat 34 • Degree of Freedom (DOF) of a rigid body is the number of independent movement it has. • e.g. 6 DOF movements
- 35. Rotation • All four components of independently results in 4 DOFs. • The constraints in 1st rule, each removes a DOF • 𝑚 + 𝑚 = 1 and 𝑚 + 𝑚 = 1 • The constraint in 2nd rule removes another DOF • 𝑚 𝑚 + 𝑚 𝑚 = 0 • 3rd rule does not remove any DOF, it eliminates half of the possible transformations. • Result is only one DOF parametrized by the angle . • All possible 2D rotation can be interpreted as points along a unit circle. Prepared by Mrs. Rashmi Bhat 35
- 36. Rotation • 3D Case of Rotation • Extension of the matrix from A from 2D to 3D results in 9 components • Start with 9 DOFs and determine what matrices remain valid rotations. Prepared by Mrs. Rashmi Bhat 36 ………(Eq. 5)
- 37. Rotation Prepared by Mrs. Rashmi Bhat 37
- 38. Rotation • 3D Case of Rotation • Columns must have unit length. • E.g. 𝑚 + 𝑚 + 𝑚 = 1 • Components of each column must lie on a unit sphere • These rules remove 3 DOFs. (remaining DOFs = 6) • Pairwise inner products of each columns must be zero • e.g. by choosing first two columns, constraint is 𝑚 𝑚 + 𝑚 𝑚 + 𝑚 𝑚 = 0 • Apply the rule to remaining pairs. • Each these cases eliminates a DOF, resulting in remaining 3DOFs only. • Avoiding mirror images, det 𝑀 = 1 does not eliminate any DOF. Prepared by Mrs. Rashmi Bhat 38
- 39. Rotation • 3D Case of Rotation • There are 3 degrees of rotational freedom. • So it is possible to pick three independent parameters fro 3D rotation and then derive all 9 components from them. • Parameterizing 3D rotaions by constructing them from “2D-like” transformations. Prepared by Mrs. Rashmi Bhat 39
- 40. 3DOF of Rotation Prepared by Mrs. Rashmi Bhat 40
- 41. Rotation Yaw, Pitch and Roll • Rotation about -axis • Let be a counterclockwise rotation of about the 𝑧 axis. • Rotation matrix is given by • Upper left of matrix looks like 2D rotation matrix. • This causes roll to behave like 2D rotation in the 𝑥𝑦 plane. Prepared by Mrs. Rashmi Bhat 41 ………(Eq. 6)
- 42. Rotation Yaw, Pitch and Roll • Rotation about -axis • Let be a counterclockwise rotation of about the axis. • Rotation matrix is given by • Points are rotated with respect to and while the coordinate is left unchanged. Prepared by Mrs. Rashmi Bhat 42 ………(Eq. 7)
- 43. Rotation Yaw, Pitch and Roll • Rotation about -axis • Let be a counterclockwise rotation of about the axis. • Rotation matrix is given by • Points are rotated with respect to and while the coordinate is left unchanged. Prepared by Mrs. Rashmi Bhat 43 ………(Eq. 6)
- 44. Rotation • Combining Rotations • The yaw, pitch, and roll rotations can be combined sequentially to attain any possible 3D rotation • The ranges of and are from to ; however, the pitch need only range from to Prepared by Mrs. Rashmi Bhat 44
- 45. Rotation • Matrix multiplications are “backwards” • Lets have a rotating point and two rotation matrices and • If we rotate using , then • If we then apply , we get • Suppose, we combine two rotations first and then apply it to to get • i.e. will give you different result • Linear algebra reads right to left • So we can combine first to obtain • Prepared by Mrs. Rashmi Bhat 45
- 46. Rotation • Translation and Rotation in one Matrix • Suppose we want to apply a rotation matrix , and follow it with a translation by 𝑥 𝑦 𝑧 = 𝑅 𝑥 𝑦 𝑧 + 𝑥 𝑦 𝑧 • There is no way to form a single 3 by 3 matrix to accomplish both operations • Homogeneous transformation matrix can be used 𝑅 𝑥 𝑦 𝑧 0 0 0 1 𝑥 𝑦 𝑧 1 = 𝑥 𝑦 𝑧 1 Prepared by Mrs. Rashmi Bhat 46
- 47. Rotation • Inverting transforms • To invert a translation, we apply the negation 𝑥 𝑦 𝑧 = 𝑅 𝑥 𝑦 𝑧 + 𝑥 𝑦 𝑧 • To invert rotation, we apply matrix inverse • For rotation matrix 𝑅, the inverse is equal to the transpose 𝑅 = 𝑅 • Inverse of is 𝑅 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 1 𝑥 𝑦 𝑧 0 0 0 1 Prepared by Mrs. Rashmi Bhat 47
- 48. Axis Angle Representation of Rotation • 3D rotation is complicated for several reasons • Nine matrix entries are specified in terms of only three independent parameters, and with no simple parameterization, • The axis of rotation is not the same every time, • The operations are non commutative, implying that the order of matrices is crucial. • Kinetic singularities Prepared by Mrs. Rashmi Bhat 48