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Planes Required Reading

## Table of Contents

```   Planes Required Reading
Abstract
Introduction
References
Plane Data Type Description
Plane routines
Constructing planes
Construct a plane from a normal vector and constant
Construct a plane from a normal vector and a point
Construct a plane from a point and spanning vectors
Access plane data elements
Examples
Converting between representations of planes
Translating planes
Applying linear transformations to planes
Finding the limb of an ellipsoid
Header examples
Use of ellipses with planes
Summary of routines
Appendix: Document Revision History
2012 JAN 23, EDW (JPL)
2008 JAN 17, BVS (JPL)
2002 DEC 12, NAIF (JPL)

```

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# Planes Required Reading

Last revised on 2012 JAN 23 by E. D. Wright.

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## Abstract

Icy contains a substantial set of subroutines that solve common mathematical problems involving planes.

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## Introduction

In Icy, 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.

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### References

1. `Calculus, Vol. II'. Tom Apostol. John Wiley and Sons, 1969. See Chapter 5, `Eigenvalues of Operators Acting on Euclidean Spaces'.

3. Icy required reading (icy.req).

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## 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 > = C
```
holds for all vectors X in the plane. C is the distance of the plane from the origin. The vector

```   C * N
```
is 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 Icy.

NAIF implemented the SPICE plane data type in IDL as a structure with the fields

```      normal:     (3-array double)
constant:   (scalar double)
```
'normal' contains the unit normal vector N; 'constant' contains the plane constant C.

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## Plane routines

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### Constructing planes

The Icy routines that create SPICE planes from various forms of data that define geometric planes:

cspice_nvc2pl

Normal vector and constant to plane
cspice_nvp2pl

Normal vector and point to plane
cspice_psv2pl

Point and spanning vectors to plane
Icy 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 Icy 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 cspice_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 Icy routines that create planes are described below. For examples of how you might use these routines in a program, see the Examples section.

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### Construct a plane from a normal vector and constant

Let `n' represent a vector normal to a plane, and `c', a scalar constant.

After `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 cspice_nvc2pl:

```   cspice_nvc2pl, n, c, plane
```

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### 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.

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 cspice_nvp2pl:

```   cspice_nvp2pl, n, p, plane
```

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### 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.

After `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 cspice_psv2pl:

```   cspice_psv2pl, p, v1, v2, plane
```

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### 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 Icy routines that break planes apart into data that define geometric planes are

cspice_pl2nvc

Plane to normal vector and constant
cspice_pl2nvp

Plane to normal vector and point
cspice_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 cspice_pl2nvc:

```   cspice_pl2nvc, plane, n, c
```
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 cspice_pl2nvp:

```   cspice_pl2nvp, plane, n, p
```
`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 cspice_pl2psv:

```   cspice_pl2psv, plane, p, v1, v2
```
`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

```   cspice_pl2nvc, n,      c,   plane
cspice_pl2nvc, plane,  n,   c

cspice_nvp2pl, p,      n,   plane
cspice_pl2nvp, plane,  p,   n

cspice_psv2pl, v1,    v2,  p,    plane
cspice_pl2psv, plane, v1,  v2,   p
```
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 cspice_pl2nvc, cspice_pl2nvp, and cspice_pl2psv 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.

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## Examples

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### Converting between representations of planes

The Icy 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 fragment

```   cspice_nvc2pl, n,      c,  plane
cspice_pl2nvp, plane,  n,  point
```

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### Translating planes

A `translation' T is a vector space mapping defined by the relation

```   T(X) = X + A   for all vectors X
```
where 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 = p + x
cspice_nvp2pl, n,     p, plane
cspice_pl2nvp, plane, n, p
```
Now, `p' is the closest point in the translated plane to the origin.

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### 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

-1
M   Y

is a point in the original plane, so

-1
< N, M  Y >  =  C.

But

-1           T  -1
< N, M  Y >  =    N  M   Y

-1 T     T
=   (  ( M  )  N  )   Y

-1 T
=   <  ( M  )  N,  Y >

So

-1 T
( M  )  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.

```   cspice_nvc2pl, n,      c,      plane
cspice_pl2psv, plane,  point,  v1,    v2
```
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.

```   cspice_mxv, m, point, tpoint
cspice_mxv, m, v1,    tv1
cspice_mxv, m, v2,    tv2
```
Construct a new SPICE plane `tplane' from the transformed point and spanning vectors, and find a unit normal and constant for this new plane.

```   cspice_psv2pl, tpoint,   tv1,  tv2,  tplane
cspice_pl2nvc, tplane,   tn,   tc
```

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### Finding the limb of an ellipsoid

This problem is somewhat artificial, because the SPICE routine cspice_edlimb already solves this problem. Nonetheless, it is a good illustration of how Icy 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 = ( p1, p2, p3 ).
```
Then every point

```   X = ( x1, x2, x3 )
```
on the limb satisfies

```   < P - X, N > = 0
```
where N a surface normal vector at X. In particular, the gradient vector

```         2      2      2
( x1/A , x2/B , x3/C  )
```
is a surface normal, so X satisfies

```   0 = < P - X, N >

2      2      2
= < P - X, (x1/A , x2/B , x3/C ) >

2      2      2                  2      2      2
= < P, (x1/A , x2/B , x3/C ) >  -  < X, (x1/A , x2/B , x3/C ) >

2      2      2
= < (p1/A , p2/B , p3/C ), X >  -  1
```
So the limb plane has normal vector

```             2      2      2
N = ( p1/A , p2/B , p3/C  )
```
and constant 1. We can create a SPICE plane representing the limb with the code fragment

```   n = [ p/a^2, p/b^2, p/c^2 ]

cspice_nvc2pl, n, 1.d, plane
```
The limb is the intersection of the limb plane and the ellipsoid. To find the intersection, we use the Icy routine cspice_inedpl (Intersection of ellipsoid and plane):

```   cspice_inedpl, a, b, c, plane, ellipse, found
```
`ellips' is a SPICE `ellipse', a data type analogous to the SPICE plane. We can use the Icy routine cspice_el2cgv (Ellipse to center and generating vectors) to find the limb's center, semi-major axis, and semi-minor axis:

```   cspice_el2cgv, ellips, center, smajor, sminor
```

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### Header examples

The headers of the plane routines (see planes.req) list additional ellipse usage examples.

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### Use of ellipses with planes

The nature of geometry problems involving planes often includes use of the SPICE ellipse data type. The example code listed in the headers of the routines cspice_inelpl and cspice_pjelpl show examples of problems solved using both the ellipse and plane data type.

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# Summary of routines

The following table summarizes the Icy plane routines.

```   cspice_inedpl        Intersection of ellipsoid and plane
cspice_inelpl        Intersection of ellipse and plane
cspice_inrypl        Intersection of ray and plane
cspice_nvc2pl        Normal vector and constant to plane
cspice_nvp2pl        Normal vector and point to plane
cspice_pjelpl        Project ellipse onto plane
cspice_pl2nvc        Plane to normal vector and constant)
cspice_pl2nvp        Plane to normal vector and point
cspice_pl2psv        Plane to point and spanning vectors
cspice_psv2pl        Point and spanning vectors to plane
cspice_vprjp         Vector projection onto plane
cspice_vprjpi        Vector projection onto plane, inverted
```

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# Appendix: Document Revision History

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### 2012 JAN 23, EDW (JPL)

Added descriptions and examples for CSPICE, Icy, and Mice distributions. Rewrote and restructured document sections for clarity and to conform to NAIF documentation standard.

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Previous edits

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### 2002 DEC 12, NAIF (JPL)

Corrections were made to comments in code example that computes altitude of ray above the limb of an ellipsoid. Previously, the quantity computed was incorrectly described as the altitude of a ray above an ellipsoid.