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find-a-curve |
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Once I had the plan to write a recipe to find a curve, given its shape.
At this moment this is a bit too ambitious, therefore I confine myself to some general remarks about form aspects.
The following questions can help to find a curve, given its form:
1. Is the curve composed of several separate curves? If so, the curve is a combined curve, take the parts apart for analysis.2. Does the curve consist of straight lines only? If so, go to the line.
3. Is the curve open, closed, simple or nonsimple?
4. How many components has the curve?
According to Harnack's curve theorem, an algebraic curve of degree D has as maximum number of components the amount of (D-1)(D-2)/2 + 1.
Examples:
- line and conic: maximum 1 component
- cubic: maximum 2 components
- quartic: maximum 4 components
- sextic: maximum 11 components
A curve that has the maximum number of components, belonging to its degree, is called an M-curve1).
1) M from maximum.