ABSTRACT
Conic sections have long been used in design. It is sometimes necessary to have an exact circle or part of a circle, while at other times only a smooth curve is necessary. The circle, ellipse, parabola, hyperbola, and the degenerate forms of lines and points give a variety of forms. Further, since, as we shall show, the general conic has five degrees of freedom, a quartic explicit polynomial would be necessary to fit a portion of the curve. A conic section has many easily found representations and it is easy to compute points along a curve with relatively fast, accurate methods. Perhaps more important than the capability to compute the particular curve is the fact that conic sections have many geometric properties which are desirable in design. One such property is that a conic can have no inflection points. Thus, one can be certain that no extraneous inflection points will be introduced when using conics, so designed curves will not have unintentional ripples. Unfortunately they also have limitations. If higher numbers of degrees of freedom are desired, piecewise methods must be used. One cannot have higher order continuity than, usually https://www.w3.org/1998/Math/MathML"> C ( 1 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429064289/fc4a4622-76a5-4c96-97e4-82b5015d6fd6/content/eq1179.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , or rarely https://www.w3.org/1998/Math/MathML"> C ( 2 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429064289/fc4a4622-76a5-4c96-97e4-82b5015d6fd6/content/eq1180.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . The lack of inflection points can be just as much a hinderance as an aid in certain applications. Also, the conic section is intrinsically a planar curve which means many pieces must be used to approximate a space curve. Despite all of these deficiencies, they have been used widely and continue to be used. For these reasons one should have an understanding of them. (a) Circle centered at <italic>C</italic>; (b) the ellipse. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429064289/fc4a4622-76a5-4c96-97e4-82b5015d6fd6/content/fig3_1.jpg" xmlns:xlink="https://www.w3.org/1999/xlink"/>
