1.1.2 Space curves

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1.1.2 Space curves

The parametric representation of space curves is:

$\displaystyle x = x(t),\;\;\;y= y(t), \;\;\;z=z(t),\;\;\;t_1\leq t \leq t_2\;.$

(1.7)

The implicit representation for a space curve can be expressed as an intersection curve between two implicit surfaces

$\displaystyle f(x,y,z)=0\;\cap\;g(x,y,z)=0\;,$

(1.8)

or parametric and implicit surfaces

$\displaystyle {\bf r}={\bf r}(u,v)\;\cap\;f(x,y,z)=0\;,$

(1.9)

or two parametric surfaces

$\displaystyle {\bf r}={\bf p}(\sigma, t)\;\cap\;{\bf r}={\bf q}(u, v)\;.$

(1.10)

The differential geometry properties of the intersection curves between implicit surfaces are discussed in Sects. 2.2 and 2.3 as well as in Chap. 6 together with the intersection curves between parametric and implicit, and two parametric surfaces. In Sect. 5.8 algorithms for computing the intersections (1.8), (1.9) and (1.10) are discussed.

If can be expressed as a function of , , or , we can eliminate from the parametric form (1.7) to generate the explicit form. Let us assume is a function of , then we have

$\displaystyle y = Y(x),\;\;\;z =Z(x)\;.$

(1.11)

This is always possible at least locally when $\frac{dx}{dt}\neq 0$ [412]. Also if the two implicit equations

and

can be solved for two of the variables in terms of the third, for example

and

in terms of

, we obtain the explicit form (1.11). This is always possible at least locally when $\frac{\partial f}{\partial y}\frac{\partial g}{\partial z} - \frac{\partial f}{\partial z}\frac{\partial g}{\partial y} \neq 0$ [412]. Therefore the explicit equation for the space curve can be expressed as an intersection curve of two cylinders projecting the curve onto

and

planes.

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December 2009