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surfpv_c

 Procedure Abstract Required_Reading Keywords Brief_I/O Detailed_Input Detailed_Output Parameters Exceptions Files Particulars Examples Restrictions Literature_References Author_and_Institution Version Index_Entries

#### Procedure

```   surfpv_c ( Surface point and velocity )

void surfpv_c ( ConstSpiceDouble      stvrtx[6],
ConstSpiceDouble      stdir [6],
SpiceDouble           a,
SpiceDouble           b,
SpiceDouble           c,
SpiceDouble           stx   [6],
SpiceBoolean        * found      )

```

#### Abstract

```   Find the state (position and velocity) of the surface intercept
defined by a specified ray, ray velocity, and ellipsoid.
```

```   None.
```

#### Keywords

```   ELLIPSOID
GEOMETRY

```

#### Brief_I/O

```   VARIABLE  I/O  DESCRIPTION
--------  ---  --------------------------------------------------
stvrtx     I   State of ray's vertex.
stdir      I   State of ray's direction vector.
a          I   Length of ellipsoid semi-axis along the X-axis.
b          I   Length of ellipsoid semi-axis along the Y-axis.
c          I   Length of ellipsoid semi-axis along the Z-axis.
stx        O   State of surface intercept.
found      O   Flag indicating whether intercept state was found.
```

#### Detailed_Input

```   stvrtx      is the state of a ray's vertex. The first three
components of `stvrtx' are the vertex's x, y, and z
position components; the vertex's x, y, and z
velocity components follow.

The reference frame relative to which `stvrtx' is
specified has axes aligned with with those of a
triaxial ellipsoid. See the description below of
the arguments `a', `b', and `c'.

The vertex may be inside or outside of this
ellipsoid, but not on it, since the surface
intercept is a discontinuous function at
vertices on the ellipsoid's surface.

and time, but these units must be consistent with
those of the other inputs.

stdir       is the state of the input ray's direction vector.
The first three components of `stdir' are a non-zero
vector giving the x, y, and z components of the
ray's direction; the direction vector's x, y, and
z velocity components follow.

`stdir' is specified relative to the same reference
frame as is `stvrtx'.

a,
b,
c           are, respectively, the lengths of a triaxial
ellipsoid's semi-axes lying along the x, y, and
z axes of the reference frame relative to which
`stvrtx' and `stdir' are specified.
```

#### Detailed_Output

```   stx         is the state of the intercept of the input ray on
the surface of the input ellipsoid. The first
three components of `stx' are the intercept's x, y,
and z position components; the intercept's x, y,
and z velocity components follow.

`stx' is specified relative to the same reference
frame as are `stvrtx' and `stdir'.

`stx' is defined if and only if both the intercept
and its velocity are computable, as indicated by
the output argument `found'.

The position units of `stx' are the same as those of
`stvrtx', `stdir', and `a', `b', and `c'. The time units are
the same as those of `stvrtx' and `stdir'.

found       is a logical flag indicating whether `stx' is
defined. `found' is SPICETRUE if and only if both the
intercept and its velocity are computable. Note
that in some cases the intercept may computable
while the velocity is not; this can happen for
near-tangency cases.
```

#### Parameters

```   None.
```

#### Exceptions

```   1)  If the input ray's direction vector is the zero vector, an
error is signaled by a routine in the call tree of this
routine.

2)  If any of the ellipsoid's axis lengths is nonpositive, an
error is signaled by a routine in the call tree of this
routine.

3)  If the vertex of the ray is on the ellipsoid, the error
SPICE(INVALIDVERTEX) is signaled by a routine in the call tree
of this routine.
```

#### Files

```   None.
```

#### Particulars

```   The position and velocity of the ray's vertex as well as the
ray's direction vector and velocity vary with time. The
inputs to surfpv_c may be considered the values of these
vector functions at a particular time, say t0. Thus

State of vertex:           stvrtx = ( V(t0), V'(t0) )

State of direction vector: stdir  = ( D(t0), D'(t0) )

To determine the intercept point, W(t0), we simply compute the
intersection of the ray originating at V(t0) in the direction of
D(t0) with the ellipsoid

2      2      2
x      y      z
--- +  --- +  ---  =  1
2      2      2
A      B      C

W(t) is the path of the intercept point along the surface of
the ellipsoid. To determine the velocity of the intercept point,
we need to take the time derivative of W(t), and evaluate it at
t0. Unfortunately W(t) is a complicated expression, and its
derivative is even more complicated.

However, we know that the derivative of W(t) at t0, W'(t0), is
tangent to W(t) at t0. Thus W'(t0) lies in the plane that is tangent
to the ellipsoid at t0. Let X(t) be the curve in the tangent plane
that represents the intersection of the ray emanating from V(t0)
with direction D(t0) with that tangent plane.

X'(t0) = W'(t0)

The expression for X'(t) is much simpler than that of W'(t);
surfpv_c evaluates X'(t) at t0.

Derivation of X(t) and X'(t)
----------------------------------------------------------------

W(t0) is the intercept point. Let N be a surface normal at I(t0).
Then the tangent plane at W(t0) is the set of points X(t) such
that

< X(t) - I(t0), N > = 0

X(t) can be expressed as the vector sum of the vertex
and some scalar multiple of the direction vector,

X(t) = V(t) + s(t) * D(t)

where s(t) is a scalar function of time. The derivative of
X(t) is given by

X'(t) = V'(t)  +  s(t) * D'(t)  +  s'(t) * D(t)

We have V(t0), V'(t0), D(t0), D'(t0), W(t0), and N, but to
evaluate X'(t0), we need s(t0) and s'(t0). We derive an
expression for s(t) as follows.

Because X(t) is in the tangent plane, it must satisfy

< X(t) - W(t0), N > = 0.

Substituting the expression for X(t) into the equation above
gives

< V(t) + s(t) * D(t) - W(t0), N > = 0.

Thus

< V(t) - W(t0), N >  +  s(t) * < D(t), N > = 0,

and
< V(t) - W(t0), N >
s(t)  =  -  ---------------------
< D(t), N >

The derivative of s(t) is given by

s'(t) =

< D(t),N > * < V'(t),N >  -  < V(t)-W(t0),N > * < D'(t),N >
-  -------------------------------------------------------------
2
< D(t), N >
```

#### Examples

```   The numerical results shown for this example may differ across
platforms. The results depend on the SPICE kernels used as input,
the compiler and supporting libraries, and the machine specific
arithmetic implementation.

1) Illustrate the role of the ray vertex velocity and
ray direction vector velocity via several simple cases. Also
show the results of a near-tangency computation.

Example code begins here.

/.
Program surfpv_ex1
./
#include <stdio.h>
#include "SpiceUsr.h"

int main()
{

SpiceBoolean            found;

SpiceDouble             a;
SpiceDouble             b;
SpiceDouble             c;
SpiceDouble             stvrtx [6];
SpiceDouble             stdir  [6];
SpiceDouble             stx    [6];

SpiceInt                i;

a = 1.0;
b = 2.0;
c = 3.0;

"     a = %f\n"
"     b = %f\n"
"     c = %f\n",
a,
b,
c                      );

for ( i = 0;  i < 3;  i++ )
{
if ( i == 0 )
{
printf ( "\n%s\n\n",
"Case 1: Vertex varies, direction is constant"  );

stvrtx[0] =  2.0;
stvrtx[1] =  0.0;
stvrtx[2] =  0.0;
stvrtx[3] =  0.0;
stvrtx[4] =  0.0;
stvrtx[5] =  3.0;

stdir[0]  = -1.0;
stdir[1]  =  0.0;
stdir[2]  =  0.0;
stdir[3]  =  0.0;
stdir[4]  =  0.0;
stdir[5]  =  0.0;
}
else if ( i == 1 )
{
printf ( "\n%s\n\n",
"Case 2: Vertex and direction both vary"  );

stvrtx[0] =  2.0;
stvrtx[1] =  0.0;
stvrtx[2] =  0.0;
stvrtx[3] =  0.0;
stvrtx[4] =  0.0;
stvrtx[5] =  3.0;

stdir[0]  = -1.0;
stdir[1]  =  0.0;
stdir[2]  =  0.0;
stdir[3]  =  0.0;
stdir[4]  =  0.0;
stdir[5]  =  4.0;
}
else
{
printf ( "\n%s\n\n",
"Case 3: Vertex and direction both vary; "
"near-tangent case"                       );

stvrtx[2] =  c - 1.e-15;
stvrtx[5] =  1.e299;
stdir[5]  =  1.e299;
}

printf ( "Vertex:\n"
" %20.12e %20.12e %20.12e\n",
stvrtx[0],
stvrtx[1],
stvrtx[2]                    );

printf ( "Vertex velocity:\n"
" %20.12e %20.12e %20.12e\n",
stvrtx[3],
stvrtx[4],
stvrtx[5]                    );

printf ( "Direction:\n"
" %20.12e %20.12e %20.12e\n",
stdir[0],
stdir[1],
stdir[2]                     );

printf ( "Direction velocity:\n"
" %20.12e %20.12e %20.12e\n",
stdir[3],
stdir[4],
stdir[5]                     );

surfpv_c ( stvrtx, stdir, a, b, c, stx, &found );

if ( !found)
{
printf ( " No intercept state found.\n" );
}
else
{
printf ( "Intercept:\n"
" %20.12e %20.12e %20.12e\n",
stx[0],
stx[1],
stx[2]                        );

printf ( "Intercept velocity:\n"
" %20.12e %20.12e %20.12e\n\n",
stx[3],
stx[4],
stx[5]                        );
}
}

return ( 0 );
}

When this program was executed on a Mac/Intel/cc/64-bit
platform, the output was:

a = 1.000000
b = 2.000000
c = 3.000000

Case 1: Vertex varies, direction is constant

Vertex:
2.000000000000e+00   0.000000000000e+00   0.000000000000e+00
Vertex velocity:
0.000000000000e+00   0.000000000000e+00   3.000000000000e+00
Direction:
-1.000000000000e+00   0.000000000000e+00   0.000000000000e+00
Direction velocity:
0.000000000000e+00   0.000000000000e+00   0.000000000000e+00
Intercept:
1.000000000000e+00   0.000000000000e+00   0.000000000000e+00
Intercept velocity:
0.000000000000e+00   0.000000000000e+00   3.000000000000e+00

Case 2: Vertex and direction both vary

Vertex:
2.000000000000e+00   0.000000000000e+00   0.000000000000e+00
Vertex velocity:
0.000000000000e+00   0.000000000000e+00   3.000000000000e+00
Direction:
-1.000000000000e+00   0.000000000000e+00   0.000000000000e+00
Direction velocity:
0.000000000000e+00   0.000000000000e+00   4.000000000000e+00
Intercept:
1.000000000000e+00   0.000000000000e+00   0.000000000000e+00
Intercept velocity:
0.000000000000e+00   0.000000000000e+00   7.000000000000e+00

Case 3: Vertex and direction both vary; near-tangent case

Vertex:
2.000000000000e+00   0.000000000000e+00   3.000000000000e+00
Vertex velocity:
0.000000000000e+00   0.000000000000e+00  1.000000000000e+299
Direction:
-1.000000000000e+00   0.000000000000e+00   0.000000000000e+00
Direction velocity:
0.000000000000e+00   0.000000000000e+00  1.000000000000e+299
Intercept:
2.580956827952e-08   0.000000000000e+00   3.000000000000e+00
Intercept velocity:
-3.874532036208e+306   0.000000000000e+00  2.999999974190e+299
```

#### Restrictions

```   None.
```

#### Literature_References

```   None.
```

#### Author_and_Institution

```   N.J. Bachman        (JPL)
J. Diaz del Rio     (ODC Space)
J.E. McLean         (JPL)
W.L. Taber          (JPL)
```

#### Version

```   -CSPICE Version 1.0.2, 05-AUG-2021 (JDR)

Edited the header to comply with NAIF standard.
Reformatted example's output to comply with maximum line length

Updated -Exceptions section.

-CSPICE Version 1.0.1, 22-JAN-2009 (NJB) (JEM) (WLT)

```   ellipsoid surface point and velocity
`Fri Dec 31 18:41:13 2021`