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 algebraic 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:

  • 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 point parameters. This problem becomes even more difficult when trying to describe
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.

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

basic physical properties of spatial parametric
curves: curvature, torsion, and the Frenet Frame.

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