TABLE OF CONTENTS The properties of a curve fall into either one of two categories: local or global. The local properties include the principal vectors (tangent, normal, and binormal), the principal planes (normal, osculating, and rectifying), curvature, and torsion. Because local properties vary from point to point on a curve, we compute them only at specific points. The global properties of a curve include arc length and whether or not it is a space curve, plane curve, or a straight line. We can apply tests to determine some of these global properties. Other properties include the presence or absence of inflection points on curves and the location of extreme or min/max points. First, we consider the local properties of a curve, expressed in terms of vectors, matrices, and determinants at a point p i on a curve. Local and global properties of a surface are the second major subject of this chapter, beginning with an introduction to some elementary, but necessary, differential geometry of surfaces. This chapter concludes with a discussion of the global geometric properties of solids, relational properties, and intersections.
We now consider in more detail some local properties of a curve. As usual, we formulate these properties in terms of vectors, matrices, and determinants. We are concerned here with properties at a point p i on a curve. At any point on a curve, we can construct a set of three orthogonal vectors; together, they are called the moving trihedron. This...
