## Geometric spatial path planning

### Implicit form

[latexpage]An implicit * planar* curve is defined as the zero of a bivariate function $f(x, y) = 0$. An

*curve is simply the case where the function $f(x, y)$ is a polynomial in $x$ and $y$ with scaling coefficients $a_{ij}$, that is $f(x, y) = \sum a_{ij}x^iy^j$. Two simple, but important, properties of implicit planar curves:*

**algebraic****Curvature:**it’s basically a measure of how much the curve is “bending”. In the limit, when the curvature is infinite, the curve becomes a*cusp***Inflection points**: An inflection point is closely related to the curvature and occurs whenever.**the sign of curvature changes**

Unlike the planar curve, a ** spatial** curve in implicit form is defined as the intersection of two implicit surfaces $f(x, y, z) \cap g(x, y, z)$.

Algebraic curves defined in implicit forms can provide a good mathematical understanding of the curve. However, they have several disadvantages in terms of curve generation as it can be * difficult to describe a motion along the curves* in terms of their

*. This problem becomes even more difficult when trying to describe*

**point parameters**spatial motions. For this reason, implicit curves are often converted to

*.*

**parametric form**### Standard parametric planar curve

A planar parametric curve involves defining the x and y coordinates with respect to some independent parameters over a certain range. Implicit curves are often converted into a parametric form for the purposes of rendering or defining a motion along the curve. curve must have a genus of 0 for a rational parameterization to exist.

$x=f(u), y=f(u), u\in [a, b]$