Today s Agenda. Geometry

Transcription

1 Today s Agenda Geometry

2 Geometry Three basic elements Points Scalars Vectors Three type of spaces (Linear) vector space: scalars and vectors Affine space: vector space + points Euclidean space: vector space + distance

3 Euclidean Spaces Distance (scalar) + a vector space Operations Vector-vector addition Scalar-vector multiplication Scalar-scalar operations Inner (dot) products v v > 0 if v 0 Magnitude (length) of a vector v = Distance between two points P & Q v v P Q = (P Q) (P Q) Angle between two vectors v & u cos θ = u v u v

4 Line Segments If we use two points to define v, then P( α) = Q + αv = Q + α (R Q) = αr + (1 α)q For 0 α 1 we get all the points on the line segment joining R and Q E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley 2012.

5 Affine Sums and Convex Hull Consider the sum P = α 1 P 1 + α 2 P α n P n Can show by induction that this sum makes sense iff α 1 + α α n = 1 in which case we have the affine sum of the points P 1,P 2,..P n If, in addition, α i >=0, we have the convex hull of P 1,P 2,..P n P 2 P 3 Smallest convex object containing P 1,P 2,..P n P 1 P E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley P 6 P 5 P 4

6 Planes A plane can be defined by three non-collinear points S α = αp + 1 α Q, 0 α 1 convex sum of P and Q convex sum of S(α) and R T β = βs + 1 β R 0 β 1 T α, β = β αα + 1 α Q + 1 β R 0 α, β 1 E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley 2012.

7 Planes A plane can also be defined by a point and two vectors P u v R T(α, β) = P + β(1 α)(q P) + (1 β)(r P) T(α, β ) = P + α u + β v E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley Q Parametric form of planes

8 Triangles A triangle can be defined by three nonclinear points Triangle is convex so any point inside can be represented as an affine sum T(α, β) = ααp + β(1 α)q + (1 β)r Affine sum T(α, β, ) = ααp + β Q + R where 0 α, β, 1 α + β + = 1 E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley 2012.

9 Barycentric Coordinates P T(a 1, a 2, a 3 )=a 1 P+a 2 Q+a 3 R where a 1 +a 2 +a 3 = 1 a i >=0 R T Q The representation is called the barycentric coordinate representation of P Where is T if a i <0? Where is T if a 1 +a 2 +a 3 1? T is outside of the triangle, but on the same plane formed by P, Q, and R T is not on the same plane PQR E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley 2012.

10 Barycentric Coordinates Equilateral triangle right triangle

11 Barycentric Coordinates Ray-triangle intersection: testing if a ray intersects with a triangle Rendering a triangle courses.cs.washington.edu

12 Normals Every plane has a vector n normal (perpendicular) to it From point-two vector form T(α, β) = P + αu + βv, we know we can use the cross product to find n = u v and the equivalent form (T P) n = 0 v P u E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley 2012.

13 3D Primitives So far, we have discussed how to represent 2D geometric objects, the primitives How about primitives in 3D? For existing graphics hardware and software, three features describe the 3D objects 2D surfaces vertices Flat, convex polygons, especially triangles

14 Reference Frame and Coordinate Systems Until now we have been able to work with geometric entities without using any frame of reference, i.e., in a coordinate-free system Coordinate system Reference frame E. Angel and D. Shriener: Interactive Computer Graphics 6E Addison-Wesley 2012

15 Coordinate Systems Consider a basis v 1, v 2,., v n of R n A vector in R n is written v = α 1 v 1 + α 2 v α n v n The list of scalars {α 1, α 2,. α n } is the representation of v with respect to the given basis We can write the representation as a row or column array of scalars α1 a = [α 1 α 2 α n] T = α2. α n E. Angel and D. Shriener: Interactive Computer Graphics 6E Addison-Wesley 2012

16 Example v=2v 1 +3v 2-4v 3 a = [2 3 4] T Note that this representation is with respect to a particular basis E. Angel and D. Shriener: Interactive Computer Graphics 6E Addison-Wesley 2012

17 Coordinate Systems Which is correct? v v Both, because coordinate system is defined in vector space and vectors have no fixed location E. Angel and D. Shriener: Interactive Computer Graphics 6E Addison-Wesley 2012

18 Frame of Reference 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 A coordinate system is insufficient to represent points Adding a reference point (origin) to a coordinate system Frame defined in affine space Frames used in graphics World frame Camera frame Image frame E. Angel and D. Shriener: Interactive Computer Graphics 6E Addison-Wesley 2012

19 Representation in a Frame Frame determined by (P 0, v 1, v 2, v 3 ) Within this frame, every vector can be written as v=α 1 v 1 + α 2 v 2 +.+α n v n Every point can be written as P = P 0 + β 1 v 1 + β 2 v 2 +.+β n v n e.g., a 3-tuples E. Angel and D. Shriener: Interactive Computer Graphics 6E Addison-Wesley 2012

20 Confusing Points and Vectors Consider the point and the vector P = P 0 + β 1 v 1 + β 2 v 2 +.+β n v n v=α 1 v 1 + α 2 v 2 +.+α n v n They appear to have the similar representations p=[β 1 β 2 β 3 ] v=[α 1 α 2 α 3 ] v p which confuses the point with the vector v Vector can be placed anywhere E. Angel and D. Shriener: Interactive Computer Graphics 6E Addison-Wesley 2012 point: fixed

21 A Single Representation - Homogeneous Coordinates If we define 0 P = 0 and 1 P =P then we can write v=α 1 v 1 + α 2 v 2 +α 3 v 3 = [α 1 α 2 α 3 0 ] [v 1 v 2 v 3 P 0 ] T P = P 0 + β 1 v 1 + β 2 v 2 +β 3 v 3 = [β 1 β 2 β 3 1 ] [v 1 v 2 v 3 P 0 ] T Thus we obtain the four-dimensional homogeneous coordinate representation v = [α 1 α 2 α 3 0 ] T p = [β 1 β 2 β 3 1 ] T E. Angel and D. Shriener: Interactive Computer Graphics 6E Addison-Wesley 2012

22 Homogeneous Coordinates In general, the homogeneous coordinates for a 3D point [x y z] is given as P = xx yy zz w T = [ww ww ww w] T When w 0, we return to a 3D point by P = [x y z 1] where x x /w, y y /w, z z /w If w=0, the representation is that of a vector E. Angel and D. Shriener: Interactive Computer Graphics 6E Addison-Wesley 2012

23 Homogeneous Coordinates and Computer Graphics Homogeneous coordinates are key to all computer graphics systems All standard transformations (rotation, translation, scaling) can be implemented with matrix multiplications using 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 E. Angel and D. Shriener: Interactive Computer Graphics 6E Addison-Wesley 2012

24 Change of Coordinate Systems Consider two representations of the same vector d with respect to two different bases. The representations are where a=[α 1 α 2 α 3 ] b=[β 1 β 2 β 3 ] d=α 1 v 1 + α 2 v 2 +α 3 v 3 = [v 1 v 2 v 3 ] [α 1 α 2 α 3 ] T =β 1 u 1 + β 2 u 2 +β 3 u 3 = [u 1 u 2 u 3 ] [β 1 β 2 β 3 ] T Let V=[v 1 v 2 v 3 ] and U=[u 1 u 2 u 3 ] d = Va = Ub

25 Representing second basis in terms of first Each of the basis vectors, u1,u2, u3, are vectors that can be represented in terms of the first basis u 1 = 11 v v v 3 u 2 = 21 v v v 3 u 3 = 31 v v v 3 d E. Angel and D. Shriener: Interactive Computer Graphics 6E Addison-Wesley 2012

26 Changing Representations The coefficients define a 3 x 3 matrix M and the bases can be related by We can change the representation from one coordinate system to the other as a=m T b = U=VM T and b=(m T ) -1 a

27 Change of Frames We can apply a similar process in homogeneous coordinates to the representations of both points and vectors Consider two frames: (P 0, v 1, v 2, v 3 ) (Q 0, u 1, u 2, u 3 ) P 0 v 1 v 3 Any point or vector can be represented in either frame We can represent Q 0, u 1, u 2, u 3 in terms of P 0, v 1, v 2, v 3 v 2 Q 0 E. Angel and D. Shriener: Interactive Computer Graphics 6E Addison-Wesley 2012 u 1 u 2 u 3

28 Representing One Frame in Terms of the Other Extending what we did with change of bases u 1 = 11 v v v 3 u 2 = 21 v v v 3 u 3 = 31 v v v 3 Q 0 = 41 v v v 3 +P 0 defining a 4 x 4 matrix = M U Q 0 = V P 0 M T E. Angel and D. Shriener: Interactive Computer Graphics 6E Addison-Wesley 2012

29 Changing Representations Any point or vector has a representation in a frame a=[α 1 α 2 α 3 α 4 ] in the first frame b=[β 1 β 2 β 3 β 4 ] in the second frame where α 4 = β 4 = 1 for points and α 4 = β 4 = 0 for vectors We can change the representation from one frame to the other as a=m T b and b=(m T ) -1 a The matrix M is 4 x 4 and specifies an affine transformation in homogeneous coordinates E. Angel and D. Shriener: Interactive Computer Graphics 6E Addison-Wesley 2012