Planes Required Reading
This required reading document is reproduced from the original NAIF document available at https://naif.jpl.nasa.gov/pub/naif/misc/toolkit_docs_N0067/C/req/planes.html
Note
These required readings documents were translated from documentation for N67 CSPICE. These pages may not be updated as frequently as the CSPICE version, and so may be out of date. Please consult the changelog for more information.
Important
NOTE any functions postfixed by "_" mentioned below are Fortan-SPICE functions unavailable in SpiceyPy as the NAIF does not officially support these with "_c" function wrappers within the CSPICE API. If these functions are necessary for your work please contact the NAIF to request that they be added to the CSPICE API
Abstract
CSPICE contains a substantial set of subroutines that solve common mathematical problems involving planes.
Introduction
In CSPICE, the 'plane' is a data representation describing planes in three-dimensional space. The purpose of the plane data type is to simplify the calling sequences of some geometry routines. Also, using a "plane" data type helps to centralize error checking and facilitate conversion between different representations of planes.
References
'Calculus, Vol. II'. Tom Apostol. John Wiley and Sons, * See Chapter 5, 'Eigenvalues of Operators Acting on Euclidean Spaces'.
Ellipses required reading (ellipses).
Plane Data Type Description
NAIF defines a SPICE plane using a unit vector N, normal to the plane, and a scalar constant C. Let< X, Y >denote the inner product of the vectors X and Y, then the relationship
< X, N > = Cholds for all vectors X in the plane. C is the distance of the plane from the origin. The vector
C * Nis the closest point in the plane to the origin. For planes that do not contain the origin, the vector N points from the origin toward the plane. The internal design of the plane data type is not part of its specification. The design is an implementation choice based on the programming language and so the design may change. Users should not write code based on the current implementation; such code might fail when used with a future version of CSPICE.
NAIF implemented the SPICE plane data type in C as a structure with the fields
SpiceDouble normal [3]; SpiceDouble constant;'normal' contains the unit normal vector N; 'constant' contains the plane constant C.
In SpiceyPy, this structure is defined by
Plane, although users are encouraged not to directly interact with this object and to instead use the spice routines described below for creating and using them.
Plane routines
Constructing planes
The CSPICE routines that create SPICE planes from various forms of data that define geometric planes:
nvc2pl()Normal vector and constant to plane
nvp2pl()Normal vector and point to plane
psv2pl()Point and spanning vectors to plane
SPICE routines that take planes as input arguments can accept planes created by any of the routines listed above. The information stored in SPICE planes is not necessarily the input information you supply to a plane-making routine. SPICE planes use a single, uniform internal representation for planes, no matter what data you use to create them. As a consequence, when you create a SPICE plane and then break it apart into data that define a plane, the returned data will not necessarily be the data you originally supplied, though they define the same geometric plane as the data you originally supplied.
This 'loss of information' may seem to be a liability at first but turns out to be a convenience in the end: the SPICE routines that break apart SPICE planes into various representations return outputs that are particularly useful for many geometric computations. In the case of the routine
pl2nvp()(Plane to normal vector and point), the output normal vector is always a unit vector, and the output point is always the closest point in the plane to the origin. The normal vector points from the origin toward the plane, if the plane does not contain the origin.You can convert any of the following representations of planes to a SPICE plane:
- A normal vector and a constant
If N is a normal vector and C is a constant, then the plane is the set of points X such that
< X, N > = C.
- A normal vector and a point
If P is a point in the plane and N is a normal vector, then the plane is the set of points X such that
< X - P, N > = 0.
- A point and two spanning vectors
If P is a point in the plane and V1 and V2 are two linearly independent but not necessarily orthogonal vectors, then the plane is the set of points
P + s * V1 + t * V2,where s and t are real numbers.
The calling sequences of the SPICE routines that create planes are described below. For examples of how you might use these routines in a program, see the Examples section.
Construct a plane from a normal vector and constant
Let 'n' represent a vector normal to a plane, and 'c', a scalar constant.Let 'n' and 'c' be declared by
n = [1.0, 1.0, 1.0] # can also use numpy arrays here c = 2.0After 'n' and 'c' have been assigned values, you can construct a SPICE plane that represents the plane having normal 'n' and constant 'c' by calling
nvc2pl():import spiceypy plane = spiceypy.nvc2pl( n, c )
Construct a plane from a normal vector and a point
Let 'n' represent a vector normal to a plane, and 'p', a point on the plane.Declare 'n' and 'p' as:
n = [1.0, 1.0, 1.0] # can also use numpy arrays here p = [1.0, 4.0, 9.0]After 'n' and 'p' have been assigned values, you can construct a SPICE plane that represents the plane containing point 'p' and having normal 'n' by calling
nvp2pl():plane = spiceypy.nvp2pl( n, p )
Construct a plane from a point and spanning vectors
Let 'p' represent a point on a plane, 'v1' and 'v2', two vectors in the plane.Let 'p', 'v1', and 'v2' be declared by
point = [..., ..., ...] # need 3 floating point numbers span1 = [..., ..., ...] # need 3 floating point numbers span2 = [..., ..., ...] # need 3 floating point numbersAfter 'p', 'v1', and 'v2' have been assigned values, you can construct a SPICE plane that represents the plane spanned by the vectors V1 and V2 and containing the point P by calling
psv2pl():plane = spiceypy.psv2pl ( p, v1, v2 )
Access plane data elements
You can 'take planes apart' as well as put them together. Any SPICE plane, regardless of which routine created it, can be converted to any of the representations listed in the previous section: normal vector and constant, point and normal vector, or point and spanning vectors.The CSPICE routines that break planes apart into data that define geometric planes are
pl2nvc()Plane to normal vector and constant
pl2nvp()Plane to normal vector and point
pl2psv()Plane to point and spanning vectors
In the following discussion, 'plane' is a SPICE plane, 'n' is a normal vector, 'p' is a point, 'c' is a scalar constant, and V1 and V2 are spanning vectors. We omit the declarations; all are as in the previous section. To find a unit normal vector 'n' and a plane constant 'c' that define 'plane', use
pl2nvc():n, c = spiceypy.pl2nvc( plane )
The constant 'c' is the distance of the plane from the origin. The vector
C * N
will be the closest point in the plane to the origin. To find a unit normal vector 'n' and a point 'p' that define 'plane', use
pl2nvp():n, p = spiceypy.pl2nvp( plane )
'p' will be the closest point in the plane to the origin. The unit normal vector 'n' will point from the origin toward the plane. To find a point 'p' and two spanning vectors 'v1' and 'v2' that define 'plane', use
pl2psv():p, v1, v2 = spiceypy.pl2psv( plane )
'p' will be the closest point in the plane to the origin. The vectors 'v1' and 'v2' are mutually orthogonal unit vectors and are also orthogonal to 'p'. It is important to note that the xxx2PL and PL2xxx routines are not exact inverses of each other. The pairs of calls
plane = spiceypy.nvc2pl( n, c ) n, c = spiceypy.pl2nvc( plane ) plane = spiceypy.nvp2pl( n, p ) n, p = spiceypy.pl2nvp( plane ) plane = spiceypy.psv2pl( p, v1, v2 ) p, v1, v2 = spiceypy.pl2psv( plane )
do not necessarily preserve the input arguments supplied to the xxx2PL routines. This is because the uniform internal representation of SPICE planes causes them to 'forget' what data they were created from; all sets of data that define the same geometric plane have the same internal representation in SPICE planes. In general, the routines
pl2nvc(),pl2nvp(), andpl2psv()are used in routines that accept planes as input arguments. In this role, they simplify the routines that call them, because the calling routines no longer check the input planes' validity.
Examples
Converting between representations of planes
The SPICE plane routines can also be used as a convenient way to convert one representation of a plane to another. For example, suppose that given a normal vector 'n' and constant 'c' defining a plane, you must produce the closest point in the plane to the origin. The code fragmentplane = spiceypy.nvc2pl( n, c ) n, point = spiceypy.pl2nvp( plane )
Translating planes
A 'translation' T is a vector space mapping defined by the relationT(X) = X + A for all vectors Xwhere A is a constant vector. While it's not difficult to directly apply a translation map to a plane, using SPICE plane routines provides the convenience of automatically computing the closest point to the origin in the translated plane. Suppose a plane is defined by the point 'p' and the normal vector 'n', and you wish to translate it by the vector 'x'. That is, you wish to find data defining the plane that results from adding 'x' to every vector in the original plane. You can do this with the code fragment
p = spiceypy.vadd( p, x ) #(Vector addition, can be done with numpy instead) plane = spiceypy.nvp2pl( n, p ) n, p = spiceypy.pl2nvp( plane )Now, 'p' is the closest point in the translated plane to the origin.
Applying linear transformations to planes
Suppose we have a normal vector N and constant C defining a plane, and we wish to apply a non-singular linear transformation T to the plane. We want to find a unit normal vector and constant that define the transformed plane; the constant should be the distance of the plane from the origin.Let T be represented by the matrix M.
If Y is a point in the transformed plane, then
\[M^{-1} Y\]is a point in the original plane, so
\[\langle N, M^{-1} Y \rangle = C.\]But
\[ \begin{align}\begin{aligned}\langle N, M^{-1} Y \rangle = N^T M^{-1} Y\\= ( ( M^{-1} )^T N )^T Y\\= \langle ( M^{-1} )^T N , Y \rangle\end{aligned}\end{align} \]So
\[( M^{-1} )^T N, C\]are, respectively, a normal vector and constant for the transformed plane.
We can solve the problem using the following code fragments. Make a SPICE plane from 'n' and 'c', and then find a point in 'plane' and spanning vectors for 'plane'. 'n' need not be a unit vector.
plane = spiceypy.nvc2pl( n, c ) point, v1, v2 = spiceypy.pl2psv( plane )Apply the linear transformation to the point and spanning vectors. All we need to do is multiply these vectors by M, since for any linear transformation T,
T ( POINT + t1 * V1 + t2 * V2 ) = T (POINT) + t1 * T (V1) + t2 * T (V2)which means that T(POINT), T(V1), and T(V2) are a a point and spanning vectors for the transformed plane.
tpoint = spiceypy.mxv( m, point ) tv1 = spiceypy.mxv( m, v1 ) tv2 = spiceypy.mxv( m, v2 )Construct a new SPICE plane 'tplane' from the transformed point and spanning vectors, and find a unit normal and constant for this new plane.
tplane = spiceypy.psv2pl( tpoint, tv1, tv2 ) tn, tc = spiceypy.pl2nvc( tplane )
Finding the limb of an ellipsoid
This problem is somewhat artificial, because the SPICE routineedlimb()already solves this problem. Nonetheless, it is a good illustration of how CSPICE plane routines are used.We'll work in body-fixed coordinates, which is to say that the ellipsoid is centered at the origin and has axes aligned with the x, y and z axes. Suppose that the semi-axes of the ellipsoid has lengths A, B, and C, and call our observation point
\[P = (p_1, p_2, p_3).\]Then every point:
\[X = (x_1, x_2, x_3)\]on the limb satisfies:
\[\langle P - X, N \rangle = 0,\]where N is a surface normal vector at X. In particular, the gradient vector:
\[\left( \frac{x_1}{A^2}, \frac{x_2}{B^2}, \frac{x_3}{C^2} \right)\]is a surface normal, so X satisfies:
\[0 = \langle P - X, N \rangle\]\[= \langle P - X, ( \frac{x_1}{A^2}, \frac{x_2}{B^2}, \frac{x_3}{C^2} ) \rangle\]\[= \langle P, ( \frac{x_1}{A^2}, \frac{x_2}{B^2}, \frac{x_3}{C^2} ) \rangle - \langle X, ( \frac{x_1}{A^2}, \frac{x_2}{B^2}, \frac{x_3}{C^2} ) \rangle\]\[= \langle ( \frac{p_1}{A^2}, \frac{p_2}{B^2}, \frac{p_3}{C^2} ), X \rangle - 1\]and constant 1. We can create a SPICE plane representing the limb with the code fragment
n[0] = p[0] / a**2 n[1] = p[1] / b**2 n[2] = p[2] / c**2 plane = spiceypy.nvc2pl( n, 1. )The limb is the intersection of the limb plane and the ellipsoid. To find the intersection, we use the CSPICE routine
inedpl()(Intersection of ellipsoid and plane):plane, ellips = spiceypy.inedpl( a, b, c )'ellips' is a SPICE 'ellipse', a data type analogous to the SPICE plane. We can use the SPICE routine
el2cgv()(Ellipse to center and generating vectors) to find the limb's center, semi-major axis, and semi-minor axis:center, smajor, sminor = spiceypy.el2cgv( ellips )
Use of ellipses with planes
Summary of routines
The following table summarizes the CSPICE plane routines.
inedpl()Intersection of ellipsoid and plane
inelpl()Intersection of ellipse and plane
inrypl()Intersection of ray and plane
nvc2pl()Normal vector and constant to plane
nvp2pl()Normal vector and point to plane
pjelpl()Project ellipse onto plane
pl2nvc()Plane to normal vector and constant
pl2nvp()Plane to normal vector and point
pl2psv()Plane to point and spanning vectors
psv2pl()Point and spanning vectors to plane
vprjp()Vector projection onto plane
vprjpi()Vector projection onto plane, inverted