ch2 - GMCh2 12/30/99 2-1 GEOMETRIC MODELING: A First Course...

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12/30/99 2-1 GEOMETRIC MODELING: A First Course Copyright © 1995-1999 by Aristides A. G. Requicha Permission is hereby granted to copy this document for individual student use at USC, provided that this notice is included in each copy. 2. Motions and Projections 2.1 Points and Vectors Imagine a small solid object and let its dimensions decrease indefinitely. The result of this conceptual experiment is modeled by a mathematical abstraction called a point . Modern mathematics defines rigorously Euclidean spaces as sets whose elements, called points , satisfy certain axioms. In everyday language we talk of “being at a point in space”, and in geometric modeling we use Euclidean points to define mathematically the locations of objects. In addition, sets of points serve to model more complicated objects, from trajectories to physical solids. Consider now a solid object in straight-line motion. The object’s velocity has a direction and a magnitude, or speed, measured e.g. in meters per second. Velocities and other physical entities such as forces that can be characterized by a direction and a magnitude are modeled mathematically by vectors . Vectors may be added by using the familiar parallelogram rule of analytical geometry and elementary mechanics. They may also be multiplied by scalar numbers. Scalar multiplication changes the magnitude of the vector but not its direction. In modern mathematics a vector space is a set of elements, called vectors, with two operations defined on them—vector addition and multiplication by a scalar—that have certain algebraic properties defined axiomatically. The vector-space axioms ensure that the usual Cartesian vectors of analytic geometry are a special case of abstract vectors. Interestingly, there are many other useful entities that are abstract vectors as well. Examples include polynomials of degree n , the spline functions we will discuss later in this course, periodic functions with period T , continuous functions in a closed interval [ a , b ], and so on. The theory of vector spaces applies equally well to all of these entities. This is a good example of the power and elegance of abstraction in modern mathematics. Points and vectors are intimately connected. In principle there are no privileged points or directions in space, i.e., space is homogeneous and isotropic. But let us pick some arbitrary point o and call it the origin . (Typically, the origin is selected for convenience in solving a specific problem.) Now each point p o plus o define a direction and a length. Therefore, for a fixed origin o , each point p corresponds to a vector x , and conversely. That is, there is a one-to-one correspondence between points and vectors. We are arguing intuitively, but the argument can be rigorized by defining point difference as an operation that produces a vector from two point arguments such that p o = x . We use the notation p o ← → x to signify that point p corresponds to vector
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ch2 - GMCh2 12/30/99 2-1 GEOMETRIC MODELING: A First Course...

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