Last week. Machiraju/Zhang/Möller

Transcription

1 Last week Machiraju/Zhang/Möller 1

2 Overview of a graphics system Output device Input devices Image formed and stored in frame buffer Machiraju/Zhang/Möller 2

3 Introduction to CG Torsten Möller 3

4 Ray tracing: the algorithm for each pixel on screen determine ray from eye through pixel if ray shoots into infinity, return a background color if ray shoots into light source, return light color appropriately find closest intersection of ray with an object cast off shadow ray (towards light sources) if shadow ray hits a light source, compute light contribution according to some illumination model cast reflected and refracted ray, recursively calculate pixel color contribution return pixel color after some absorption Most expensive Machiraju/Zhang/Möller 4

5 The z-buffer algorithm operations vs. per-pixel for ray tracing Simple and often accelerated with hardware Works regardless of the order in which the polygons are processed no need to sort them back to front A visibility algorithm and not designed to compute colors WebGL implements this fundamental algorithm gl.enable(gl.depth_test); gl.clear(gl.color_buffer_bit gl.depth_buffer_bit); Machiraju/Zhang/Möller 5

6 The algorithm for each polygon in the scene project its vertices onto viewing (image) plane end if for each pixel inside the polygon formed on viewing plane determine point on polygon corresponding to this pixel get pixel color according to some illumination model get depth value for this pixel (distance from point to plane) if depth value < stored depth value for the pixel end if update pixel color in frame buffer update depth value in depth buffer (z-buffer) Question: What does the depth buffer store after running the z-buffer algorithm for the scene? Does polygon order change depth? Image? Machiraju/Zhang/Möller 6

7 Geometric Basics Introduction to Computer Graphics Torsten Möller Machiraju/Zhang/Möller

8 Graphics Pipeline Hardware Modelling Transform Visibility Illumination + Shading Perception, Interaction Texture/ Color Realism Machiraju/Zhang/Möller 8

9 Reading Chapter 4 of Angel Chapter 7 of Foley, van Dam, Machiraju/Zhang/Möller 9

10 Schedule Geometry basics Affine transformations Use of homogeneous coordinates Concatenation of transformations 3D transformations Transformation of coordinate systems Transform the transforms Transformations in OpenGL Machiraju/Zhang/Möller 10

11 Why do we need transformations? Need to transform objects from object coordinate system (OCS) to world CS Need to transform objects from world CS to the coordinate system of the camera (ViewCS) Need to transform objects from VCS into an WebGL window/viewport (DisplayCS) Object may move (translate or rotate) or deform (scale, shear, or general non-rigid deformation) WCS VCS DCS OCS Machiraju/Zhang/Möller 11

12 Geometry basics Scalar, point, and vector Vector space and affine space Basic point and vector operations Sided-ness test Lines, planes, and triangles Linear independence Coordinate systems and frames Machiraju/Zhang/Möller 13

13 Scalar, point, and vector Point: a location in space Specified by a k-tuple for k-d points Always given with respect to some coordinate system Scalar: a quantity, e.g., edge length Vector: a directed line segment between points Spaces: vector space, affine space, Euclidean space, etc. P = x y z Machiraju/Zhang/Möller 14

14 Vector space A set of vectors with scalar multiplications and vector additions Scalar-vector multiplication u = αv Vector-vector addition: w = u + v Expressions such as v = u + 2w 3r make sense in a vector space But vectors lack position Inadequate for representing geometry we need positions, which are given by points Machiraju/Zhang/Möller 15

15 Affine space A vector space + points = affine space Operations Vector-vector addition Scalar-vector multiplication Point-vector addition Affine sum of points and convex sums A vector space + distance/norm = Euclidean space Machiraju/Zhang/Möller 16

16 Geometry basics Scalar, point, and vector Vector space and affine space Basic point and vector operations Sided-ness test Lines, planes, and triangles Linear independence Coordinate systems and frames Machiraju/Zhang/Möller 17

17 Basic point and vector operations point point = vector point + vector = point vector operations: scalar * vector = vector vector + vector = vector vector vector = scalar, the dot product vector vector = vector, the cross product u v v u Right-hand rule Machiraju/Zhang/Möller 18

18 More on dot product u v = ux vx + uy vy + uz vz u u = u 2 is always non-negative u v is commutative and distributive over additions u v = u v cos θ Two vectors u and v are orthogonal if and only if u v = 0 If v is normalized, i.e., v = 1, then u v gives the projection of u in the direction of v Machiraju/Zhang/Möller 19

19 More on cross product Cross product u v is a vector perpendicular to u and v frequently used to compute the normal to a plane Direction of the cross product is determined by the right hand rule u v = - v u Wikipedia u v = u v sin θ = area of the parallelogram u v v u Wikipedia Machiraju/Zhang/Möller 20

20 More on cross product How to compute? use determinant i j k u1 u2 u3 v1 v2 v3 Wikipedia Machiraju/Zhang/Möller 21

21 Affine and convex sums Addition of two arbitrary points is not defined in an affine space But consider two points P and Q with Q = P + αv, we can always find a point R such that v = R P so now we have Q = P + α(r P) or Q = αr + (1 α)p Thus, affine sum (combination) of points can be defined t1p1 + t2p2 + + tnpn, t1 + t2 + + tn = 1 Convex sum (combination) of points t1p1 + + tnpn, where t1 + + tn = 1 and ti 0 for all i Machiraju/Zhang/Möller 22

22 Convex hull Convex hull of a set of points: set of convex combination of these points Alternatively, the convex hull is the smallest convex object containing the set of points Formed by shrink wrapping points Useful in, e.g., fast collision detection Machiraju/Zhang/Möller 23

23 Sided-ness test On which side does a point V lie with respect to a line, specified by a vector u? Solution 1: use an implicit line or plane equation; plug in the point coordinates and check the sign Solution 2: in 2D, let the z coordinate be zero, compute a cross product and check the sign of the z component Solution 3: find a vector u perpendicular to u; check sign of u v ( u y, u x ) u = (u x, u y ) V u V V u u V Machiraju/Zhang/Möller 24

24 Geometry basics Scalar, point, and vector Vector space and affine space Basic point and vector operations Sided-ness test Lines, planes, and triangles Linear independence Coordinate systems and frames Machiraju/Zhang/Möller 25

25 Line representations Consider all points of the form P(α) = P 0 + αd this is the set of all points lying on a line that passes through P 0 in the direction of the vector d Known as the parametric form of the line Given two points R and Q on the line, we have Other representations Explicit: y = mx + h Implicit: ax + by + c = 0 x(α) = α R x + (1 α) Q x y(α) = α R y + (1 α) Q y Machiraju/Zhang/Möller 26

26 Ray and line segment If α 0, then P(α) is the ray leaving P 0 in the direction of the vector d For the two-point representation x(α) = α R x + (1 α) Q x Ԧv y(α) = α R y + (1 α) Q y if 1 α 0, then we get all the points on the line segment joining R and Q Machiraju/Zhang/Möller 27

27 Parametric plane representation A plane can be defined by a point and two vectors or by three points P Ԧv P P (a, b) (a, b) R u R Q P(a, b) = R + a u + b Ԧv P(a, b) = R + a(q R) + b(p R) Ԧv u Machiraju/Zhang/Möller 28

28 Implicit plane equation F(x, y, z) = ax + by + cz + d = 0 Typically, (a, b, c) is the unit normal of the plane If F(x 0, y 0, z 0 ) < 0, point (x 0, y 0, z 0 ) is below the plane (w. r. t. the normal) (a, b, c) If F(x 0, y 0, z 0 ) > 0, point (x 0, y 0, z 0 ) is above the plane In general, the distance from a point (x 0, y 0, z 0 ) to the plane is given by d Machiraju/Zhang/Möller 29

29 Triangle and barycentric coordinates convex sum of P and Q convex sum of S(α) & R T is a convex sum of P, Q, and R. The weights are called the barycentric coodinates of T Machiraju/Zhang/Möller 30

30 Coordinate system (CS) and frame n linearly independent vectors of an n-d vector space define a coordinate system (CS), e.g., Cartesian CS The vectors are called the basis vectors Any vector in the space can be written as a linear sum of the basis vectors in a unique way An origin O, the reference point, along with a set of basis vectors form a frame Any point P = O + linear sum of basis vectors Coefficients of the sum: coordinates of point P Machiraju/Zhang/Möller 31

31 From one CS to another Express point given in one frame/cs in another CS In 2D, need to solve system of two equations Assume the two CS have the same origin P = (1, 2) T P = (x, y) T Basis: (1, 1) T and (0, 1) T New set of basis: (-1,1) T and (1, 0) T Need to find x and y such that 1*(1,1) T + 2*(0,1) T = x*(2,3) T + y*( 1, 2) T Alternatively (in matrix notation) Machiraju/Zhang/Möller 32

32 Change of CS This reduces to a change of basis vectors With the 2D basis vectors u (u ) and v (v ), find a 2 2 matrix M such that [u v] M = [u v ], thus M = [u v] 1 [u v ] Then P(u, v ) = M 1 P(u, v) for all points P The matrix transforms points from one CS, with basis u and v, to another CS, with basis u and v M is also called the change of basis matrix Machiraju/Zhang/Möller 33

33 Next Lecture: 2-dimensional transformations Machiraju/Zhang/Möller