shape

1.Curves and surfaces

We describe the type of curves and surfaces provided in the plugin SemiAlgebraic of the package shape for axel.

1.1.Rational curves

A rational curve is defined as the set of points

where is the dimension of the ambient space and are univariate polynomials (with coefficients in ).

If , the rational curve is also called a polynomial curve.

Bézier curves are special types of rational curves, which use Bernstein bases to represent the polynomial functions.

A Bézier parametric curve is the image of a map of the form

where are the control points, and , are the Bernstein basis elements of degree for the interval .

A rational Bézier curve is constructed as follows:

where are the weights of the control points (usually ).

1.1.1.Axel format

<curve type="rational" color="0 255 0">
  <domain>0 1</domain>
  <polynomial>t</polynomial>
  <polynomial>t^2</polynomial>
  <polynomial>t^3</polynomial>
  <polynomial>1</polynomial>
</curve>

1.2.Rational surfaces

A rational curve is defined as the image of a map of the form

where is the dimension of the ambient space and are bivariate polynomials (with coefficient in ).

1.2.1.Axel format

<surface type="rational" color="0 255 0">
 <domain>0 0 1 0 0 1</domain>
  <polynomial>s</polynomial>
  <polynomial>t</polynomial>
  <polynomial>s^2+t^2</polynomial>
  <polynomial>1</polynomial>
</surface>

1.3.Algebraic curves in the plane

An algebraic curve in is defined by one equation of the form:

where is a polynomial with coefficieints in .

1.3.1.Example

Here is an image of the curve defined by the equation

1.3.2.Axel format

<curve type="algebraic" name="nice_curve_10">
  <domain>-4.1 4.2 -3.1 3</domain>
  <polynomial>-y^8+x^7-7*x^6*y+21*x^5*y^2-35*x^4*y^3+35*x^3*y^4
-21*x^2*y^5+7*x*y^6-y^7+8*y^6-7*x^5+35*x^4*y-70*x^3*y^2+70*x^2*y^3
-35*x*y^4+7*y^5-20*y^4+14*x^3-42*x^2*y+42*x*y^2-14*y^3+16*y^2
-7*x+7*y-2</polynomial>
</curve>

1.4.Algebraic curves in 3D

An algebraic curve in is defined by two of more equations of the form:

The solution set of these equations is of dimension . In other words, the tangent linear space at almost all solution points is a line.

1.4.1.Example

Here we see two surfaces intersecting in a blue curve:

1.4.2.Axel format

<curve type="algebraic" color="0 0 255">
  <domain>-3 3.1 -3 3.1 -3 3.1</domain>
  <polynomial>10000*x^4+10000*y^4+10000*z^4-40000*x^2-40000*y^2*z^2-40000*y^2
-40000*z^2*x^2-40000*z^2-40000*x^2*y^2+207846*x*y*z+10000</polynomial>
  <polynomial>40000*x^3-80000*x-80000*x*z^2-80000*x*y^2+207846*y*z</polynomial>
</curve>

1.5.Algebraic surfaces

A general implicit surface is defined as the solution set of one equation:

where is a subdomain of . Usually is continous and enough differentiable so

An algebraic surface is a special case of an implicit surface where is a polynomial with coefficients in .

1.5.1.Example

Here is an example of a surface defined by the equation:

which is known as the Whitney umbrella.

1.5.2.Axel format

<surface type="algebraic" name="whitney umbrella" color="0 170 255" >
  <domain>-3 3 -3 3 -3 3</domain>
  <polynomial variables="x y z">x^2*y-z^2</polynomial>
</surface>

2.Algorithms

© 2012 Bernard Mourrain