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Simbody
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Simbody
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This ContactGeometry subclass represents an ellipsoid centered at the origin, with its principal axes pointing along the x, y, and z axes and half dimensions a,b, and c (all > 0) along those axes, respectively. More...
#include <ContactGeometry.h>
Inheritance diagram for SimTK::ContactGeometry::Ellipsoid:Public Member Functions | |
| Ellipsoid (const Vec3 &radii) | |
| Construct an Ellipsoid given its three principal half-axis dimensions a,b,c (all positive) along the local x,y,z directions respectively. | |
| const Vec3 & | getRadii () const |
| Obtain the three half-axis dimensions a,b,c used to define this ellipsoid. | |
| void | setRadii (const Vec3 &radii) |
| Set the three half-axis dimensions a,b,c (all positive) used to define this ellipsoid, overriding the current radii and recalculating the principal curvatures at a cost of about 50 flops. | |
| const Vec3 & | getCurvatures () const |
| For efficiency we precalculate the principal curvatures whenever the ellipsoid radii are set; this avoids having to repeatedly perform these three expensive divisions at runtime. | |
| UnitVec3 | findUnitNormalAtPoint (const Vec3 &P) const |
| Given a point P =(x,y,z) on the ellipsoid surface, return the unique unit outward normal to the ellipsoid at that point. | |
| Vec3 | findPointWithThisUnitNormal (const UnitVec3 &n) const |
| Given a unit direction n, find the unique point P on the ellipsoid surface at which the outward-facing normal is n. | |
| Vec3 | findPointInSameDirection (const Vec3 &Q) const |
| Given a direction d defined by the vector Q-O for an arbitrary point in space Q=(x,y,z)!=O, find the unique point P on the ellipsoid surface that is in direction d from the ellipsoid origin O. | |
| void | findParaboloidAtPoint (const Vec3 &Q, Transform &X_EP, Vec2 &k) const |
| Given a point Q on the surface of the ellipsoid, find the approximating paraboloid at Q in a frame P where OP=Q, Pz is the outward-facing unit normal to the ellipsoid at Q, Px is the direction of maximum curvature and Py is the direction of minimum curvature. | |
| void | findParaboloidAtPointWithNormal (const Vec3 &Q, const UnitVec3 &n, Transform &X_EP, Vec2 &k) const |
| If you already have both a point and the unit normal at that point, this will save about 50 flops by trusting that you have provided the correct normal; be careful -- no one is going to check that you got this right. | |
| const EllipsoidImpl & | getImpl () const |
| Internal use only. | |
| EllipsoidImpl & | updImpl () |
| Internal use only. | |
Static Public Member Functions | |
| static bool | isInstance (const ContactGeometry &geo) |
| Return true if the supplied ContactGeometry object is an Ellipsoid. | |
| static const Ellipsoid & | getAs (const ContactGeometry &geo) |
| Cast the supplied ContactGeometry object to a const Ellipsoid. | |
| static Ellipsoid & | updAs (ContactGeometry &geo) |
| Cast the supplied ContactGeometry object to a writable Ellipsoid. | |
| static ContactGeometryTypeId | classTypeId () |
| Obtain the unique id for Ellipsoid contact geometry. | |
This ContactGeometry subclass represents an ellipsoid centered at the origin, with its principal axes pointing along the x, y, and z axes and half dimensions a,b, and c (all > 0) along those axes, respectively.
The implicit equation f(x,y,z)=0 of the ellipsoid surface is
f(x,y,z) = Ax^2+By^2+Cz^2 - 1
where A=1/a^2, B=1/b^2, C=1/c^2
A,B, and C are the squares of the principal curvatures ka=1/a, kb=1/b, and kc=1/c.
The interior of the ellipsoid consists of all points such that f(x,y,z)<0 and points exterior satisfy f(x,y,z)>0. The region around any point (x,y,z) on an ellipsoid surface is locally an elliptic paraboloid with equation
-2 z' = kmax x'^2 + kmin y'^2
where z' is measured along the the outward unit normal n at (x,y,z), x' is measured along the the unit direction u of maximum curvature, and y' is measured along the unit direction v of minimum curvature. kmax,kmin are the curvatures with kmax >= kmin > 0. The signs of the mutually perpendicular vectors u and v are chosen so that (u,v,n) forms a right-handed coordinate system for the paraboloid.
| SimTK::ContactGeometry::Ellipsoid::Ellipsoid | ( | const Vec3 & | radii | ) | [explicit] |
Construct an Ellipsoid given its three principal half-axis dimensions a,b,c (all positive) along the local x,y,z directions respectively.
The curvatures (reciprocals of radii) are precalculated here at a cost of about 50 flops.
| const Vec3& SimTK::ContactGeometry::Ellipsoid::getRadii | ( | ) | const |
Obtain the three half-axis dimensions a,b,c used to define this ellipsoid.
| void SimTK::ContactGeometry::Ellipsoid::setRadii | ( | const Vec3 & | radii | ) |
Set the three half-axis dimensions a,b,c (all positive) used to define this ellipsoid, overriding the current radii and recalculating the principal curvatures at a cost of about 50 flops.
| [in] | radii | The three half-dimensions of the ellipsoid, in the ellipsoid's local x, y, and z directions respectively. |
| const Vec3& SimTK::ContactGeometry::Ellipsoid::getCurvatures | ( | ) | const |
For efficiency we precalculate the principal curvatures whenever the ellipsoid radii are set; this avoids having to repeatedly perform these three expensive divisions at runtime.
The curvatures are ka=1/a, kb=1/b, and kc=1/c so that the ellipsoid's implicit equation can be written Ax^2+By^2+Cz^2=1, with A=ka^2, etc.
Given a point P =(x,y,z) on the ellipsoid surface, return the unique unit outward normal to the ellipsoid at that point.
If P is not on the surface, the result is the same as for the point obtained by scaling the vector P - O until it just touches the surface. That is, we compute P'=findPointInThisDirection(P) and then return the normal at P'. Cost is about 50 flops regardless of whether P was initially on the surface.
| [in] | P | A point on the ellipsoid surface, measured and expressed in the ellipsoid's local frame. See text for what happens if P is not actually on the ellipsoid surface. |
Given a unit direction n, find the unique point P on the ellipsoid surface at which the outward-facing normal is n.
Cost is about 50 flops.
| [in] | n | The unit vector for which we want to find a match on the ellipsoid surface, expressed in the ellipsoid's local frame. |
Given a direction d defined by the vector Q-O for an arbitrary point in space Q=(x,y,z)!=O, find the unique point P on the ellipsoid surface that is in direction d from the ellipsoid origin O.
That is, P=s*d for some scalar s > 0 such that f(P)=0. Cost is about 50 flops.
| [in] | Q | A point in space measured from the ellipsoid origin but not the origin. |
| void SimTK::ContactGeometry::Ellipsoid::findParaboloidAtPoint | ( | const Vec3 & | Q, |
| Transform & | X_EP, | ||
| Vec2 & | k | ||
| ) | const |
Given a point Q on the surface of the ellipsoid, find the approximating paraboloid at Q in a frame P where OP=Q, Pz is the outward-facing unit normal to the ellipsoid at Q, Px is the direction of maximum curvature and Py is the direction of minimum curvature.
k=(kmax,kmin) are the returned curvatures with kmax >= kmin > 0. The equation of the resulting paraboloid in the P frame is -2z = kmax*x^2 + kmin*y^2. Cost is about 270 flops; you can save a little time if you already know the normal at Q by using the other overloaded signature for this method.
| [in] | Q | A point on the surface of this ellipsoid, measured and expressed in the ellipsoid's local frame. |
| [out] | X_EP | The frame of the paraboloid P, measured and expressed in the ellipsoid local frame E. X_EP.p() is Q, X_EP.x() is the calculated direction of maximum curvature kmax; y() is the direction of minimum curvature kmin; z is the outward facing normal at Q. |
| [out] | k | The maximum (k[0]) and minimum (k[1]) curvatures of the ellipsoid (and paraboloid P) at point Q. |
| void SimTK::ContactGeometry::Ellipsoid::findParaboloidAtPointWithNormal | ( | const Vec3 & | Q, |
| const UnitVec3 & | n, | ||
| Transform & | X_EP, | ||
| Vec2 & | k | ||
| ) | const |
If you already have both a point and the unit normal at that point, this will save about 50 flops by trusting that you have provided the correct normal; be careful -- no one is going to check that you got this right.
The results are meaningless if the point and normal are not consistent. Cost is about 220 flops.
| static bool SimTK::ContactGeometry::Ellipsoid::isInstance | ( | const ContactGeometry & | geo | ) | [inline, static] |
Return true if the supplied ContactGeometry object is an Ellipsoid.
| static const Ellipsoid& SimTK::ContactGeometry::Ellipsoid::getAs | ( | const ContactGeometry & | geo | ) | [inline, static] |
Cast the supplied ContactGeometry object to a const Ellipsoid.
| static Ellipsoid& SimTK::ContactGeometry::Ellipsoid::updAs | ( | ContactGeometry & | geo | ) | [inline, static] |
Cast the supplied ContactGeometry object to a writable Ellipsoid.
| static ContactGeometryTypeId SimTK::ContactGeometry::Ellipsoid::classTypeId | ( | ) | [static] |
Obtain the unique id for Ellipsoid contact geometry.
| const EllipsoidImpl& SimTK::ContactGeometry::Ellipsoid::getImpl | ( | ) | const |
Internal use only.
Reimplemented from SimTK::ContactGeometry.
| EllipsoidImpl& SimTK::ContactGeometry::Ellipsoid::updImpl | ( | ) |
Internal use only.
Reimplemented from SimTK::ContactGeometry.
1.7.3