1. 1 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 Geometry

2. 2 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 Objectives Introduce the elements of geometry ­Scalars ­Vectors ­Points Define basic primitives ­Line segments ­Polygons

3. 3 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 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 a minimum set of primitives from which we can build more sophisticated objects We will need three basic elements ­Scalars ­Vectors ­Points

4. 4 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 Scalars Need three basic elements in geometry ­Scalars, Vectors, Points can be combined by two operations (addition and multiplication) Examples include the real and complex number system Scalars alone have no geometric properties

5. 5 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 Vectors Physical definition: a vector is a quantity with two attributes ­Direction ­Magnitude Examples include ­Force ­Velocity ­Directed line segments v

6. 6 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 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, undefined orientation The sum of any two vectors is a vector ­Use head-to-tail axiom v -v vv v u w

7. 7 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 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

8. 8 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 Vectors Lack Position These vectors are identical ­Same length and magnitude Vectors spaces insufficient for geometry ­Need points

9. 9 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 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

10. 10 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 Affine Spaces Point + a vector space Operations ­Vector-vector addition ­Scalar-vector multiplication ­Point-vector addition ­Scalar-scalar operations For any point define ­1 P = P ­0 P = 0 (zero vector)

11. 11 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 Lines Consider all points of the form ­P(  )=P 0 +  d ­Set of all points that pass through P 0 in the direction of the vector d

12. 12 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 Rays and Line Segments If  >= 0, then P(  ) is the ray leaving P 0 in the direction d If we use two points to define v, then P(  ) = Q +  (R-Q)=Q+  v =  R + (1-  )Q For 0<=  <=1 we get all the points on the line segment joining R and Q

13. 13 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 Convexity An object is convex iff for any two points in the object all points on the line segment between these points are also in the object P Q Q P convex not convex

14. 14 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 Affine Sums Consider the “sum” P=  1 P 1 +  2 P 2 +…..+  n P n iff  1 +  2 +…..  n =1 Then we have the affine sum of the points P 1  P 2,…..P n If, in addition,  i >=0, we have the convex hull of P 1  P 2,…..P n

15. 15 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 Convex Hull Smallest convex object containing P 1  P 2,…..P n Formed by “shrink wrapping” points

16. 16 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 Planes A plane can be defined by a point and two vectors or by three points P( ,  )=R+  u+  v P( ,  )=R+  (Q-R)+  (P-Q) u v R P R Q

17. 17 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 Triangles convex sum of P and Q convex sum of S(  ) and R for 0<= ,  <=1, we get all points in triangle

18. 18 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 Normals Every plane has a vector n normal (perpendicular, orthogonal) to it From point-two vector form P( ,  )=R+  u+  v, we know we can use the cross product to find n = u  v and the equivalent form (P(  )-P)  n=0 u v P