Convert_ParameterisationType.hxx File Reference

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: More...

Enumeration Type Documentation

Convert_ParameterisationType

enum Convert_ParameterisationType

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).

Enumerator
Convert_TgtThetaOver2
Convert_TgtThetaOver2_1
Convert_TgtThetaOver2_2
Convert_TgtThetaOver2_3
Convert_TgtThetaOver2_4
Convert_QuasiAngular
Convert_RationalC1
Convert_Polynomial