MbSilhouetteCurve Silhouette Curve

The MbSilhouetteCurve class is declared in the cur_silhouette_curve.h file.

A silhouette curve of a surface MbSilhouetteCurve is a derived class of a curve on surface MbSurfaceCurve. A silhouette curve is defined by a surface MbSurface surface, a two-dimensional curve MbCurve curve in the surface parameter space, a Boolean flag of curve periodicity closed, a Boolean flag of perspective perspective, a view vector eye, and the curve type species. The curve also contains some additional data that is not mandatory and is used to speed up the work of the curve methods.

A silhouette curve is a curve on the surface surface that separates the part of the surface visible from the observation point from the part of the surface invisible from the observation point. If perspective = true, then the observation point is defined by the view vector eye. If perspective = false, then the observation point is located at infinity and the view vector eye defines the direction from the observation point to the surface. The normal of the surface surface at the silhouette curve is orthogonal to the straight line connecting this point on the surface with the observation point.

In the particular case when an exact silhouette curve of the surface can be constructed, the curve type species takes the value cbt_Ordinary. For example, for a sphere the silhouette curve is a circle. In the particular case, an exact spatial curve exactCurve is constructed, which exactly describes the silhouette of the surface and is used to calculate the radius vector of the silhouette curve and its derivatives.

In the general case, the curve type species takes the value cbt_Specific, the two-dimensional curve curve is a spline and approximates the silhouette of the surface. In the general case, a point of the silhouette curve is calculated by an iterative method that uses the two-dimensional curve curve as an initial approximation.

The parameter domain of the silhouette curve of the surface coincides with the parameter domain of the two-dimensional curve curve.

In the PointOn( double & t, MbCartPoint3D & r ) method, the radius vector of the curve r is described by the vector function

\[\mathbf{r}(t) = \mathbf{surface}(u, v),\]

where u, v are the coordinates of a two-dimensional point whose initial approximation is calculated by the curve → PointOn( t, point ) method, u = point.x, v = point.y. Then the parameters u and v are refined by an iterative method that uses the equation

\[\mathbf{vector}\cdot\mathbf{n}(u, v) = 0,\]

where n(u, v) is the surface normal that is calculated by the surface → Normal( u, v, n ) method, and vector is the view vector (for an observation point at infinity vector = eye).

A silhouette curve of a torus surface is shown in Fig. 1:

Fig. 1

A silhouette curve of the torus surface from another direction is shown in Fig. 2:

Fig. 2

When crossing the silhouette curve, the scalar product of the surface normal vector and the view vector changes sign. A silhouette curve is always either closed or starts and ends at the surface boundaries. A silhouette curve is used to construct projections of the silhouette of a curvilinear surface onto a plane.