Focus On Curves and Surfaces (Focus on Game Development)

In Conclusion

I am moving fairly quickly because I suspect that this is review for most of the readers. If this is not the case, spend some time working with the math and plot out some curves. This will give you a good feel for the concepts and it should help you internalize the material. Here are some important things to keep in mind as you review this chapter.

• Most of my mathematical definitions are far from rigorous . There are entire books written on some of the ideas I cover in one sentence . Most of the proofs in those books are not necessary for the general practitioner. If you enjoy the material and want to know more, there are some very good books available. A few of them are described in Appendix C.

• Lines are first-degree polynomial functions. The slope of the line defines the direction and the slope is the same for all points along the line.

• Higher-degree polynomials require differentiation to determine the slope at any point along the curve. The derivative of a function is used to compute the slope for any value of the input variable.

• Continuity between two adjoining curves can be on many levels. Zero-order continuity means that the two curves are actually joined on the graph. Higher orders of continuity apply if higher derivatives of the functions are also continuous.

• The curve applications all share a common framework. The positions of the points on the curves are given in pixels, so you may need to scale accordingly when drawing higher-degree polynomials. This will become less of a burden in later chapters.

• So far, the concepts are fairly simple and atomic. Spend time with the concepts and read Appendix A if there are things you don't understand.