计算机图形学computer-graph(16)课件

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1、D,,Click to Edit Master Text Styles,,SECOND LEVEL,,THIRD LEVEL,,*,PPT&Programs,,,,,http://211.87.235.32/share/ComputerGraphics/,1,,Geometry,Shandong University Software College,,2,,Objectives,Introduce the elements of geometry,,Scalars,,Vectors,,Points,,Develop mathematical operations among them in

2、 a,coordinate-free,manner,,Define basic primitives,,Line segments,,Polygons,,3,,Geometry in Computer Graphics,,4,,Basic Elements,Geometry is the study of the relationships among objects in an n-dimensional space,,In computer graphics, we are interested in objects that exist in three dimensions,,Want

3、 a minimum set of primitives from which we can build more sophisticated objects,,We will need three basic elements,,Scalars,,Vectors,,Points,5,,Scalars,Scalars can be defined as members of sets which can be combined by two operations (addition and multiplication) obeying some fundamental axioms (ass

4、ociativity, commutivity, inverses)(,结合律、交换律、逆,),,Examples include the real and complex number systems under the ordinary rules with which we are familiar,,Scalars alone have no geometric properties,6,,Vectors,Physical definition: a vector is a quantity with two attributes,,Direction,,Magnitude,,Exam

5、ples include,,Force,,Velocity,,Directed line segments,,Most important example for graphics,,Can map to other types,v,7,,Vector Operations,Every vector has an inverse,,Same magnitude but points in opposite direction,,Every vector can be multiplied by a scalar,,There is a zero vector,,Zero magnitude,

6、undefined orientation,,The sum of any two vectors is a vector,,Use head-to-tail axiom,（三角形法则）,v,-v,,v,v,u,w,v,u,w,8,,Coordinate-Free Geometry,Cartesian approach,,Points were at locations in space,p,=(x,y,z),,We derived results by algebraic manipulations involving these coordinates,,Coordinate-Free

7、Geometry was nonphysical,,Physically, points exist regardless of the location of an arbitrary coordinate system,,Most geometric results are independent of the coordinate system,,Example Euclidean geometry: two triangles are identical if two corresponding sides and the angle between them are identica

8、l,9,,Linear Vector Spaces,Mathematical system for manipulating vectors,,Operations,,Scalar-vector multiplication,u,=,,v,,Vector-vector addition:,w,=,u,+,v,,Expressions such as,,v=u+2w-3r,,Make sense in a vector space,10,,Vectors Lack Position,These vectors are identical,,Same length and magnitude,,

9、,,,,,,,Vectors spaces insufficient for geometry,,Need points,y,,x,v,=(,x, y,),11,,Points,Location in space,,Operations allowed between points and vectors,,Point-point subtraction yields a vector,,Equivalent to point-vector addition,P=v+Q,v=P-Q,,y,,x,P,= (,x, y,),12,,Affine Spaces,Point + a vector sp

10、ace,,Operations,,Vector-vector addition -> vector,,Scalar-vector multiplication ->vector,,Point-vector addition -> point,,Scalar-scalar operations -> scalar,13,,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,14,,Para

11、metric 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,) = (1-,a),x,0,+,a,x,1,,y(,a,) = (1-,a,)y,0,+,a,y,1,,x,0,x,1,a,0,,1,,15,

12、,Affine sum,If,a,>= 0, then,P(,a,),is the,ray,leaving P,0,in the direction,d,,If we use two points to define,v,, then,,P(,a,) = Q +,a,(R-Q)=Q+,a,v,,=,a,R + (1-,a,)Q,,For,0<=,a,<=1,we get all the,,points on the,line segment,,joining,R,and,Q,16,,Convexity,An object is,convex,iff for any two points in

13、the object all points on the line segment between these points are also in the object,P,Q,Q,P,convex,not convex,S,S,17,,Affine Sums,Consider the “sum”,,P=,a,1,P,1,+,a,2,P,2,+…..+,a,n,P,n,,Can show by induction that this sum makes sense iff,,a,1,+,a,2,+…..+,a,n,=1,,in which case we have the,affine su

14、m,of the points,P,1,,,P,2,,…..P,n,,If, in addition,,a,i,>=0,, we have the,convex hull,of,P,1,,,P,2,,…..P,n,,18,,Convex Hull,Smallest convex object containing,P,1,,,P,2,,…..,P,n,,Formed by “shrink wrapping” points,19,,Dot & Cross Products,The Vector Dot Product,,Given two n-dimensional vectors [x,1,,

15、x,2,, …, x,n,]T and [y,1,,y,2,, …, y,n,]T, define their,inner product,(,dot product,) to be x,1,y,1,+ … + x,n,y,n,.,,The dot product of vectors,v,and,w,is generally denoted by,v . w,,The,length,of a vector v is . We denote this length by ||,v,||,,The distance between P and Q is the length

16、of the vector Q - P,,20,,Dot & Cross Products,Properties of the Dot Product,,Symmetric:,v,.,w = w,.,v,,Nondegenerate:,v,.,v = 0 v =0,,Bilinear:,v,.,(u+ aw) = v,.,u + a (v,.,w ),,Normalizing a vector( generate vectors whose length is 1, unit vector):,,The angle between the vectors v and w is:,,,,V

17、,U,,Ɵ,|U|cos,Ɵ,21,,Dot & Cross Products,Cross product,:,v=[v,1,,v,2,,v,3,],t,and w=[w,1,,w,2,,w,3,],t,,,,,,The vector denoted by v*w has the property that it is perpendicular to the plane defined by v and w,,,Its length is the product ||v||*||w||*|sin A| , where A is the angle between v and w.,,Righ

18、t Hand Coordinate System,22,,Curves and Surfaces,Curves are one parameter entities of the form,P,(,a,),where the function is nonlinear,,Surfaces are formed from two-parameter functions,P,(,a,,,b,),,Linear functions give planes and polygons,P(,a,),P(,a,,,b,),23,,Planes,A plane can be defined by a poi

19、nt and two vectors or by three points,P(,a,,,b,)=R+,a,u+,b,v,P(,a,,,b,)=R+,a,(Q-R)+,b,(P-R),u,v,R,P,,R,Q,,,,24,,Triangles,convex sum of P and Q,convex sum of S(,a,) and R,for 0<=,a,,,b,<=1, we get all points in triangle,25,,Normals,Every plane has a vector n normal (perpendicular, orthogonal) to it,

20、,From point-two vector form,,P(,a,,,b,)=P,0,+,a,u+,b,v,,,we know we can use the cross product to find,n,=,u,,,v,,and the equivalent form,,(P(,a,)-P,0,), n=0,,,,u,v,P,0,26,,Geometry Primitive,,27,,Linear Independence,A set of vectors,v,1,,,v,2,, …,,v,n,is,linearly independent,if,,,a,1,v,1,+,a,2,v,

21、2,+..,a,n,v,n,=0 iff,a,1,=,a,2,=…=0,,If a set of vectors is linearly independent, we cannot represent one in terms of the others,,If a set of vectors is linearly dependent, as least one can be written in terms of the others,28,,Dimension,In a vector space, the maximum number of linearly independent

22、vectors is fixed and is called the,dimension,of the space,,In an,n,-dimensional space, any set of n linearly independent vectors form a,basis,for the space,,Given a basis,v,1,,,v,2,,….,,v,n,, any vector,v,can be written as,,v=,a,1,v,1,+,a,2,v,2,,+….+,a,n,v,n,,where the {,a,i,} are unique,29,,Represe

23、ntation,Until now we have been able to work with geometric entities without using any frame of reference, such as 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 coordina

24、tes,,Camera coordinates,30,,Coordinate Systems,Consider a basis,v,1,,,v,2,,….,,v,n,,A vector is written,v=,a,1,v,1,+,a,2,v,2,,+….+,a,n,v,n,,The list of scalars,{,a,1,,,a,2,,,….,a,n,},is the,representation,of,,v,,with respect to the given basis,,We can write the representation as a row or column arra

25、y of scalars,a,=[,a,1,,a,2,,….,a,n,],T,=,31,,Example,v=2v,1,+3v,2,-4v,3,,a,=[2 3 –4],T,,Note that this representation is with respect to a particular basis,,For example, in OpenGL we start by representing vectors using the object basis but later the system needs a representation in terms of the cam

26、era or eye basis,32,,Coordinate Systems,Which is correct?,,,,,,,Both are because vectors have no fixed location,v,v,33,,Frames,A coordinate system is insufficient to represent points,,If we work in an affine space we can add a single point, the,origin,, to the basis vectors to form a,frame,O,v,1,v,2

27、,v,3,34,,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=,a,1,v,1,+,a,2,v,2,,+….+,a,n,v,n,,Every point can be written as,,P,=,P,0,+,b,1,v,1,+,b,2,v,2,,+….+,b,n,v,n,35,,Confusing Points and Vectors,Consider the point and the

28、 vector,,,P,=,P,0,+,b,1,v,1,+,b,2,v,2,,+….+,b,n,v,n,,v=,a,1,v,1,+,a,2,v,2,,+….+,a,n,v,n,,They appear to have the similar representations,,p,=[,b,1,b,2,b,3,],v,=[,a,1,a,2,a,3,],,which confuses the point with the vector,,A vector has no position,v,p,v,Vector can be placed anywhere,,point: fixed,36,,A

29、Single Representation,If we define,0•P =,0,and,1•P =P,then we can write,,,v,=,a,1,v,1,+,a,2,v,2,,+,a,3,v,3,= [,a,1,a,2,,a,3,,0,,],,[,v,1,v,2,v,3,,P,0,],T,,P,=,P,0,+,b,1,v,1,+,b,2,v,2,,+,b,3,v,3,= [,b,1,b,2,,b,3,,1,,],,[,v,1,v,2,v,3,,P,0,],T,,Thus we obtain the four-dimensional,homogeneous coordinate

30、,representation,,v,=,[,a,1,a,2,,a,3,,0,,],T,,p,=,[,b,1,b,2,,b,3,,1,,],T,37,,Homogeneous Coordinates,The homogeneous coordinates form for a three dimensional point [x y z] is given as,,p,=[x’ y’ z’ w],T,,=[wx wy wz w],T,,We return to a three dimensional point (for w,0,) by,,x,x’,/w,,y,,y’/w,,z,,

31、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,,For w=1, the representation of a point is [x y z 1],38,,Homogeneous Coordinates and Computer Graphics,Homogeneous coordinates ar

32、e 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

33、 points,,For perspective we need a,perspective division,39,,Change of Coordinate Systems,Consider two representations of a the same vector with respect to two different bases. The representations are,v,=,a,1,v,1,+,a,2,v,2,,+,a,3,v,3,= [,a,1,a,2,,a,3,],,[,v,1,v,2,v,3,],T,,=,b,1,u,1,+,b,2,u,2,,+,b,3,u

34、,3,= [,b,1,b,2,,b,3,],,[,u,1,u,2,u,3,],T,,a,=,[,a,1,,a,2,,a,3,],b,=,[,b,1,,b,2,,b,3,],where,40,,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,=,g,11,v,1,+,g,12,v,2,+,g,13,v,3,,u,2,=,g,21,v,1,+,g

35、,22,v,2,+,g,23,v,3,,u,3,=,g,31,v,1,+,g,32,v,2,+,g,33,v,3,v,41,,Matrix Form,The coefficients define a 3 x 3 matrix,,,,,,and the bases can be related by,,,,see text for numerical examples,a=M,T,b,M,=,42,,Change of Frames,We can apply a similar process in homogeneous coordinates to the representations

36、of both points and vectors,,,,,,,,,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,,Consider two frames:,,(P,0,, v,1,, v,2,, v,3,),,(Q,0,, u,1,, u,2,, u,3,),P,0,v,1,v,2,v,3,Q,0,u,1,u,2,u,3,43,,Representing One Frame in

37、Terms of the Other,u,1,=,g,11,v,1,+,g,12,v,2,+,g,13,v,3,,u,2,=,g,21,v,1,+,g,22,v,2,+,g,23,v,3,,u,3,=,g,31,v,1,+,g,32,v,2,+,g,33,v,3,,Q,0,=,g,41,v,1,+,g,42,v,2,+,g,43,v,3,+,g,44,P,0,,Extending what we did with change of bases,defining a 4 x 4 matrix,M,=,44,,Working with Representations,Within the two

38、 frames any point or vector has a representation of the same form,,w = a,1,v,1,+ a,2,v,2,+ a,3,v,3,+ a,4,P,0,,a,=,[,a,1,,a,2,,a,3,a,4,],in the first frame,,,w = a,T,v,,b,=,[,b,1,,b,2,,b,3,b,4,],in the second frame,,w = b,T,u,,,where,a,4,=,,b,4,= 1,for points and,,a,4,=,,b,4,= 0,for vectors and,,,,

39、The matrix,M,is 4 x 4 and specifies an affine transformation in homogeneous coordinates,,a=M,T,b,w=b,T,u = b,T,Mv= a,T,v,45,,Affine Transformations,Every linear transformation is equivalent to a change in frames,,Every affine transformation preserves lines,,However, an affine transformation has only

40、 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,46,,The World and Camera Frames,When we work with representations, we work with n-tuples or arrays of scalars,,Changes in frame are then defined by 4 x 4 matrices,,I

41、n OpenGL, the base frame that we start with is the world frame,,Eventually we represent entities in the camera frame by changing the world representation using the model-view matrix,,Initially these frames are the same (,M,=,I,),47,,Moving the Camera,If objects are on both sides of z=0, we must move camera frame,M,=,48,,注：,文档资料素材和资料部分来自网络，如不慎侵犯了您的权益，请联系文库客服，我们将做删除处理，感谢您的理解。,