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Chapter 4

Chapter 4. Geometric Objects and Transformation. Lines. Consider all points of the form P( a )=P 0 + a d Set of all points that pass through P 0 in the direction of the vector d. Parametric Form. This form is known as the parametric form of the line

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Chapter 4

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  1. Chapter 4 Geometric Objects and Transformation

  2. Lines • Consider all points of the form • P(a)=P0 + ad • Set of all points that pass through P0 in the direction of the vector d

  3. Parametric Form • This form is known as the parametric form of the line • More robust and general than other forms • Extends to curves and surfaces • Two-dimensional forms • Explicit: y = mx +h • Implicit: ax + by +c =0 • Parametric: x(a) = ax0 + (1-a)x1 y(a) = ay0 + (1-a)y1

  4. Rays and Line Segments If a >= 0, then P(a) is the ray leaving P0 in the direction d If we use two points to define v, then P( a) = Q + a (R-Q)=Q+av =aR + (1-a)Q For 0<=a<=1 we get all the points on the line segment joining R and Q

  5. Dot and Cross: Products

  6. Three-Dimensional Primitives 3D curves 3D surfaces Volumetric Objects Hollow objects Objects can be specified by vertices Simple and flat polygons (triangles) Constructive Solid Geometry (CSG)

  7. Constructive Solid Geometry

  8. Representation • Until now we have been able to work with geometric entities without using any frame of reference, such a coordinate system • Need a frame of reference to relate points and objects to our physical world. • For example, where is a point? Can’t answer without a reference system • World coordinates • Camera coordinates

  9. Confusing Points and Vectors v p v can place anywhere fixed Consider the point and the vector P = P0 + b1v1+ b2v2+….+bnvn v=a1v1+ a2v2+….+anvn They appear to have the similar representations P=[b1 b2 b3] v=[a1 a2 a3] which confuse the point with the vector A vector has no position

  10. A Single Representation If we define 0•P = 0 and 1•P =P then we can write v=a1v1+ a2v2+a3v3 = [a1 a2a30][v1 v2 v3 P0] T P = P0 + b1v1+ b2v2+b3v3= [b1 b2b31][v1 v2 v3 P0] T Thus we obtain the four-dimensional homogeneous coordinate representation v = [a1 a2a30] T P = [b1 b2b31] T

  11. Homogeneous Coordinates The general form of four dimensional homogeneous coordinates is p=[x y x w] T We return to a three dimensional point (for w0) by xx/w yy/w zz/w If w=0, the representation is that of a vector Note that homogeneous coordinates replaces points in three dimensions by lines through the origin in four dimensions

  12. Homogeneous Coordinates and Computer Graphics • Homogeneous coordinates are key to all computer graphics systems • All standard transformations (rotation, translation, scaling) can be implemented by matrix multiplications with 4 x 4 matrices • Hardware pipeline works with 4 dimensional representations • For orthographic viewing, we can maintain w=0 for vectors and w=1 for points • For perspective we need a perspective division

  13. Representing a Mesh e2 v5 v6 e3 e9 e8 v8 v4 e1 e11 e10 v7 e4 e7 v1 e12 v2 v3 e6 e5 • Consider a mesh • There are 8 nodes and 12 edges • 5 interior polygons • 6 interior (shared) edges • Each vertex has a location vi = (xi yi zi)

  14. Inward and Outward Facing Polygons The order {v0, v3, v2, v1} and {v1, v0, v3, v2} are equivalent in that the same polygon will be rendered by OpenGL but the order {{v0, v1, v2, v3} is different The first two describe outwardly facing polygons Use the right-hand rule = counter-clockwise encirclement of outward-pointing normal OpenGL treats inward and outward facing polygons differently

  15. Geometry versus Topology • Generally it is a good idea to look for data structures that separate the geometry from the topology • Geometry: locations of the vertices • Topology: organization of the vertices and edges • Example: a polygon is an ordered list of vertices with an edge connecting successive pairs of vertices and the last to the first • Topology holds even if geometry changes

  16. Bilinear Interpolation Assuming a linear variation, then we can make use of the same interpolation coefficients in coordinates for the interpolation of other attributes.

  17. Scan-line Interpolation A polygon is filled only when it is displayed It is filled scan line by scan line Can be used for other associated attributes with each vertex

  18. General Transformations A transformation maps points to other points and/or vectors to other vectors

  19. Linear Function (Transformation) Transformation matrix for homogeneous coordinate system:

  20. Affine Transformations – 1/2 • Line preserving • Characteristic of many physically important transformations • Rigid body transformations: rotation, translation • Scaling, shear • Importance in graphics is that we need only transform endpoints of line segments and let implementation draw line segment between the transformed endpoints

  21. Affine Transformations – 2/2 Every linear transformation (if the corresponding matrix is nonsingular) is equivalent to a change in frames However, an affine transformation has only 12 degrees of freedom because 4 of the elements in the matrix are fixed and are a subset of all possible 4 x 4 linear transformations

  22. Translation P´ d P • Move (translate, displace) a point to a new location • Displacement determined by a vector d • Three degrees of freedom • P’=P+d

  23. Rotation (2D) – 1/2 x´ = r cos (f + q) = rcosf cosq - rsinf sinq y´ = r sin (f + q) = rcosf sinq + rsinf cosq x´ =x cos q –y sin q y´ = x sin q + y cos q x = r cos f y = r sin f • Consider rotation about the origin by q degrees • radius stays the same, angle increases by q

  24. Rotation (2D) – 2/2 Using the matrix form: There is a fixed point Could be extended to 3D Positive direction of rotation is counterclockwise 2D rotation is equivalent to 3D rotation about the z-axis

  25. (Non-)Rigid-Body Transformation Non-rigid-bodytransformations Translation and rotation are rigid-body transformation

  26. Scaling x´=sxx y´=syx z´=szx Uniform and non-uniform scaling Expand or contract along each axis (fixed point of origin)

  27. Reflection sx = -1 sy = 1 original sx = -1 sy = -1 sx = 1 sy = -1 corresponds to negative scale factors

  28. Transformation in Homogeneous Coordinates With a frame, each affine transformation is represented by a 44 matrix of the form:

  29. Translation note that this expression is in four dimensions and expresses that point = vector + point Using the homogeneous coordinate representation in some frame p=[ x y z 1]T p´=[x´ y´ z´ 1]T d=[dx dy dz 0]T Hence p´= p + dor x´=x+dx y´=y+dy z´=z+dz

  30. Translation Matrix We can also express translation using a 4 x 4 matrix T in homogeneous coordinates p´=Tp where This form is better for implementation because all affine transformations can be expressed this way and multiple transformations can be concatenated together

  31. Rotation about the Z axis x´=x cos q –y sin q y´= x sin q + y cos q z´=z • Rotation about z axis in three dimensions leaves all points with the same z • Equivalent to rotation in two dimensions in planes of constant z • or in homogeneous coordinates p’=Rz(q)p

  32. Rotation Matrix

  33. Rotation about X and Y axes • Same argument as for rotation about z axis • For rotation about x axis, x is unchanged • For rotation about y axis, y is unchanged

  34. Scaling Matrix x´=sxx y´=syx z´=szx p´=Sp

  35. Inverses • Although we could compute inverse matrices by general formulas, we can use simple geometric observations • Translation: T-1(dx, dy, dz) = T(-dx, -dy, -dz) • Rotation: R-1(q) = R(-q) • Holds for any rotation matrix • Note that since cos(-q) = cos(q) and sin(-q)=-sin(q) R-1(q) = R T(q) • Scaling: S-1(sx, sy, sz) = S(1/sx, 1/sy, 1/sz)

  36. Concatenation We can form arbitrary affine transformation matrices by multiplying together rotation, translation, and scaling matrices Because the same transformation is applied to many vertices, the cost of forming a matrix M=ABCD is not significant compared to the cost of computing Mp for many vertices p The difficult part is how to form a desired transformation from the specifications in the application

  37. Order of Transformations Note that matrix on the right is the first applied Mathematically, the following are equivalent p´ = ABCp = A(B(Cp)) Note many references use column matrices to present points. In terms of column matrices p´T = pTCTBTAT

  38. Rotation about a Fixed Point and about the Z axis f 

  39. General Rotation about the Origin R(q) = Rx() Ry() Rz() y , ,are called the Euler angles v q Note that rotations do not commute We can use rotations in another order but with different angles x z A rotation by q about an arbitrary axis can be decomposed into the concatenation of rotations about the x, y, and z axes

  40. Rotation about an Arbitrary Axis – 1/2 Final rotation matrix:     Normalize u:  

  41. Rotation about an Arbitrary Axis – 2/2 Rotate the line segment to the plane of y=0, and the line segment is foreshortened to:        Rotate clockwise about the y-axis, so:   Final transformation matrix:

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