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Show m^mn^fmmmtfmmm PPIIPHII I mu ^■•wmpwm««-' HIBWIUUIMPW IBIIIII ii»i'i ■ ■"■' ■■ ■■ '"" ->■ " /5 CHAPTER THREE SUBDIVIDING A CUBIC CURVE A method for quicKly subdividing a cubic curve is presented in this chapter; the extension to patches is developed in the next chapter. The method uses a new kind of difference equation for obtaining the midpoint of a curve segment. The resulting ability to quickly subdivide a curve makes the application of the subdivision algorithm practical. SUBDIVIDING THE CUBIC CURVE Subdivision is easy because, as we shall see, the midpoint of a cubic curve is the average of its two endpoints minus a correction term. One result of this is tha' the cubic can be subdivided with only three adds. A similar method can be used to find the derivative at the midpoint. Consider the cubic: f(t) - at3 ♦ bt7 * ct ♦ d. The problem is to find f(t) when f(t+h) and f(t-h) are alreac/ known. Note first that: ^mm |
