Lecture 9: Vector Geometry: A Coordinate-Free ApproachAnd ye shall know the truth, and the truth shall make you free. John 8:321. Coordinate-Free MethodsWe are going to ascend now from the flat world of 2-dimensions into the real world of 3-dimensions. When working in 3-dimensions, we shall insist predominantly on coordinate-free methods. Even in 2-dimensions we adopted coordinate-free techniques to describe shapes because it is easier to represent geometry without troubling about coordinates. For example, it is simpler to describe a bump using a turtle program rather than trying to delineate the coordinates of all the vertices of the bump. Similarly, we invoke affine transformations -- translation, rotation, scaling, and shear -- to move and reshape geometry without worrying about the entries -- the coordinates -- of the corresponding matrices. Coordinates are useful for computations, but conceptually we prefer to work at a higher level of abstraction. Turtle programs and affine transformations were our entry to coordinate-free methods in 2-dimensions.In 3-dimensions coordinate-free methods are even more crucial. You may be comfortable with coordinate techniques in 2-dimensions, but in 3-dimensions it is much harder to conceptualize geometry using coordinates. In addition to allowing us to work at a higher level of abstraction, coordinate-free methods provide a more concise notation. Typically we will need to deal with only one equation for a point or a vector, rather than with three equations for their coordinates. Also we shall see that coordinate-free methods lead naturally to geometrically meaningful expressions. With coordinates we can perform senseless computations that have no intrinsic geometric significance -- for example, we could add the coordinates of two points. Coordinate-free methods will help us to avoid the pitfalls of such mindless computations.Finally, and most importantly, coordinate-free techniques capture the geometric meaning behind our computational methods. You may know that the dot product of two vectors u,v can be calculated from the formula€ u•v = u1v1+ u2v2+ u3v3,but why would anyone ever want to compute the expression involving the rectangular coordinates on the right hand side? The geometric meaning of the dot product is captured by the coordinate-free expression€ u•v =|u | | v | cos(ϑ),where € | u | and € | v | are the lengths of the vectors u and v, and € ϑ is the angle between u and v. Angle and length (as well as projection), these quantities are the geometric content of the dot product, content that is completely hidden in the coordinate computation.Our approach to Computer Graphics is to simplify geometry as much as possible by invoking the Mathematics most appropriate to the problem at hand. In 2-dimensions we employ Turtle Graphics (LOGO) rooted in conformal transformations and Affine Graphics (CODO) based on affine transformations. In 3-dimensions, affine transformations along with a new transformation, perspective projection, play an even more central role. These transformations are applied to manipulate shapes in 3-dimensions, but at bottom most shapes in 3-dimensions are represented in terms of points and vectors. Therefore we begin our study of 3-dimensional Computer Graphics by introducing a coordinate-free approach to the algebra and geometry of points and vectors.2. Vectors and Vector SpacesVectors and vector spaces should be familiar to you from standard courses on linear algebra. Vectors can be added, subtracted, and multiplied by scalars, and these vector operations all have coordinate-free definitions (see Figure 1). v + wwvv − wwvwcw(a) addition (b) subtraction (c) scalar multiplicationFigure 1: Coordinate-free geometric definitions of (a) addition, (b) subtraction, and (c) scalar multiplication for vectors.These vector operations obey the usual laws of arithmetic: addition is associative (Figure 2(a)) and commutative (Figure 2(b)) and scalar multiplication distributes through addition (Figure 2(c)).€ u€ w€ v€ u + v€ v + w€ u + (v + w) = (u + v) + w€ u€ u€ v€ v€ v + u = u +v€ u€ cu€ v€ cv€ u + v€ c(u + v)(a) associative (b) commutative (c) distributiveFigure 2: The associative, commutative, and distributive properties of vector addition and scalar multiplication. Vector addition is (a) associative because no matter how we group € u,v, w, if we place these vectors head to tail, the vector € u + v + w goes from the tail of u to the head of w. Vector addition is (b) commutative because € v + u, u + v both represent the diagonal of the parallelogram with sides u, v. Finally, by similar triangles, (c) scalar multiplication distributes through addition.2Nevertheless, although vectors and vector operations are useful and familiar, vectors are not the primary focus of Computer Graphics. On the graphics terminal we see points, not vectors, so it is to points that we next turn our attention.3. Points and Affine SpacesPoints are not vectors. Points have a fixed position, but no direction or length; vectors have direction and length, but no fixed location. Vectors can be added, subtracted, and multiplied by scalars and the result is always a vector. Points can be subtracted from one another, but the result is a vector, not a point (see Figure 3), A vector can be added to a point, and the result is a point (see Figure 3), but there is no coordinate-free way to add two points or to multiply a point by a scalar.P + vQ − PvP••••QPFigure 3: Subtracting a point from a point, and adding a vector to a point. Notice that € P + (Q − P) = Q, so the usual cancellation law of addition applies.Expressions of the form € ckvkk∑ are called linear combinations. For vectors, linear combinations always make sense because addition and scalar multiplication are always defined for vectors. Although we cannot, in general, add two points or multiply points by scalars, nevertheless some linear combinations of points also make sense. For example, the expression€ P + Q2=P2+Q2 represents the midpoint of the line segment joining the points P and Q, even though none of the expressions € P + Q, € P / 2, € Q