Editing Bézier curve (section)

===Properties===
[[File:quadratic_to_cubic_Bezier_curve.svg|thumb|upright|A cubic Bézier curve (yellow) can be made identical to a quadratic one (black) by
<div style="margin-left:1em;text-indent:-1em">1. copying the end points, and</div>
<div style="margin-left:1em;text-indent:-1em">2. placing its 2 middle control points (yellow circles) 2/3 along line segments from the end points to the quadratic curve's middle control point (black rectangle).</div>]]
* The curve begins at <math>\mathbf P_0</math> and ends at <math>\mathbf P_n</math>; this is the so-called ''endpoint interpolation'' property.
* The curve is a line [[if and only if]] all the control points are [[collinear]].
* The start and end of the curve is [[tangent]] to the first and last section of the Bézier polygon, respectively.
* A curve can be split at any point into two subcurves, or into arbitrarily many subcurves, each of which is also a Bézier curve.
* Some curves that seem simple, such as the [[circle]], cannot be described exactly by a Bézier or [[piecewise]] Bézier curve; though a four-piece cubic Bézier curve can approximate a circle (see [[composite Bézier curve]]), with a maximum radial error of less than one part in a thousand, when each inner control point (or offline point) is the distance <math>\textstyle\frac{4\left(\sqrt {2}-1\right)}{3}</math> horizontally or vertically from an outer control point on a unit circle. More generally, an ''n''-piece cubic Bézier curve can approximate a circle, when each inner control point is the distance <math>\textstyle\frac{4}{3}\tan(t/4)</math> from an outer control point on a unit circle, where  <math display="inline">t = 2\pi/n</math> (i.e. <math>t=360^\circ/n</math>), and <math>n>2</math>.
* Every quadratic Bézier curve is also a cubic Bézier curve, and more generally, every degree ''n'' Bézier curve is also a degree ''m'' curve for any ''m'' > ''n''. In detail, a degree ''n'' curve with control points <math>\mathbf{P}_0,\,\dots,\,\mathbf{P}_n</math> is equivalent (including the parametrization) to the degree ''n'' + 1 curve with control points <math>\mathbf{P}'_0,\,\dots,\,\mathbf {P}'_{n+1}</math>, where <math>\mathbf P'_k=\tfrac{k}{n+1}\mathbf P_{k-1}+\left(1-\tfrac{k}{n+1}\right)\mathbf P_k</math>, <math>\forall k = 0,\,1,\,\dots,\,n,\,n+1</math> and define <math>\mathbf{P}_{n+1} := \mathbf{P}_{0}</math>, <math>\mathbf{P}_{-1} := \mathbf{P}_{n}</math>.
* Bézier curves have the [[variation diminishing property]]. What this means in intuitive terms is that a Bézier curve does not "undulate" more than the polygon of its control points, and may actually "undulate" less than that.<ref name="GonzalezDiaz-Herrera2014">{{cite book |author1=Teofilo Gonzalez |author1-link=Teofilo F. Gonzalez |author2=Jorge Diaz-Herrera |author3=Allen Tucker |title=Computing Handbook, Third Edition: Computer Science and Software Engineering |url=https://books.google.com/books?id=vMqSAwAAQBAJ&pg=SA32-PA14 |year=2014 |publisher=CRC Press |isbn=978-1-4398-9852-9 |at=page 32-14<!--this is a single page number-->}}</ref>
* There is no [[local control]] in degree ''n'' Bézier curves—meaning that any change to a control point requires recalculation of and thus affects the aspect of the entire curve, "although the further that one is from the control point that was changed, the smaller is the change in the curve".<ref name="Agoston2005">{{cite book |author=Max K. Agoston |title=Computer Graphics and Geometric Modelling: Implementation & Algorithms |url=https://books.google.com/books?id=TAYw3LEs5rgC&pg=PA404 |year=2005 |publisher=Springer Science & Business Media |isbn=978-1-84628-108-2 |page=404}}</ref>
* A Bézier curve of order higher than two may intersect itself or have a [[cusp (singularity)|cusp]] for certain choices of the control points.