# Convert_ParameterisationType.hxx File Reference

`#include <Standard_PrimitiveTypes.hxx>`

## Enumerations

enum | Convert_ParameterisationType { Convert_TgtThetaOver2, Convert_TgtThetaOver2_1, Convert_TgtThetaOver2_2, Convert_TgtThetaOver2_3, Convert_TgtThetaOver2_4, Convert_QuasiAngular, Convert_RationalC1, Convert_Polynomial }

Identifies a type of parameterization of a circle or ellipse represented as a BSpline curve.
For a circle with a center C and a radius R (for example a Geom2d_Circle or a Geom_Circle),
the natural parameterization is angular. It uses the angle Theta made by the vector CM with
the 'X Axis' of the circle's local coordinate system as parameter for the current point M. The
coordinates of the point M are as follows:
X = R *cos ( Theta )
y = R * sin ( Theta )
Similarly, for an ellipse with a center C, a major radius R and a minor radius r, the circle Circ
with center C and radius R (and located in the same plane as the ellipse) lends its natural
angular parameterization to the ellipse. This is achieved by an affine transformation in the plane
of the ellipse, in the ratio r / R, about the 'X Axis' of its local coordinate system. The
coordinates of the current point M are as follows:
X = R * cos ( Theta )
y = r * sin ( Theta )
The process of converting a circle or an ellipse into a rational or non-rational BSpline curve
transforms the Theta angular parameter into a parameter t. This ensures the rational or
polynomial parameterization of the resulting BSpline curve. Several types of parametric
transformations are available.
TgtThetaOver2
The most usual method is Convert_TgtThetaOver2 where the parameter t on the BSpline
curve is obtained by means of transformation of the following type:
t = tan ( Theta / 2 )
The result of this definition is:
cos ( Theta ) = ( 1. - t**2 ) / ( 1. + t**2 )
sin ( Theta ) = 2. * t / ( 1. + t**2 )
which ensures the rational parameterization of the circle or the ellipse. However, this is not the
most suitable parameterization method where the arc of the circle or ellipse has a large opening
angle. In such cases, the curve will be represented by a BSpline with intermediate knots. Each
span, i.e. each portion of curve between two different knot values, will use parameterization of
this type.
The number of spans is calculated using the following rule:
( 1.2 * Delta / Pi ) + 1
where Delta is equal to the opening angle (in radians) of the arc of the circle (Delta is
equal to 2.* Pi in the case of a complete circle).
The resulting BSpline curve is "exact", i.e. computing any point of parameter t on the BSpline
curve gives an exact point on the circle or the ellipse.
TgtThetaOver2_N
Where N is equal to 1, 2, 3 or 4, this ensures the same type of parameterization as
Convert_TgtThetaOver2 but sets the number of spans in the resulting BSpline curve to N
rather than allowing the algorithm to make this calculation.
However, the opening angle Delta (parametric angle, given in radians) of the arc of the circle
(or of the ellipse) must comply with the following:
- Delta <= 0.9999 * Pi for the Convert_TgtThetaOver2_1 method, or
- Delta <= 1.9999 * Pi for the Convert_TgtThetaOver2_2 method.
QuasiAngular
The Convert_QuasiAngular method of parameterization uses a different type of rational
parameterization. This method ensures that the parameter t along the resulting BSpline curve is
very close to the natural parameterization angle Theta of the circle or ellipse (i.e. which uses
the functions sin ( Theta ) and cos ( Theta ).
The resulting BSpline curve is "exact", i.e. computing any point of parameter t on the BSpline
curve gives an exact point on the circle or the ellipse.
RationalC1
The Convert_RationalC1 method of parameterization uses a further type of rational
parameterization. This method ensures that the equation relating to the resulting BSpline curve
has a "C1" continuous denominator, which is not the case with the above methods. RationalC1
enhances the degree of continuity at the junction point of the different spans of the curve.
The resulting BSpline curve is "exact", i.e. computing any point of parameter t on the BSpline
curve gives an exact point on the circle or the ellipse.
Polynomial
The Convert_Polynomial method is used to produce polynomial (i.e. non-rational)
parameterization of the resulting BSpline curve with 8 poles (i.e. a polynomial degree equal to 7).
However, the result is an approximation of the circle or ellipse (i.e. computing the point of
parameter t on the BSpline curve does not give an exact point on the circle or the ellipse).
. More...

## Enumeration Type Documentation

enum Convert_ParameterisationType

Enumeration values:

  Convert_TgtThetaOver2 |
  Convert_TgtThetaOver2_1 |
  Convert_TgtThetaOver2_2 |
  Convert_TgtThetaOver2_3 |
  Convert_TgtThetaOver2_4 |
  Convert_QuasiAngular |
  Convert_RationalC1 |
  Convert_Polynomial |