Using vectors

From d54fe7c1f704a63824c5bfa0ece65245572e9b27 Mon Sep 17 00:00:00 2001 From: Joseph Hunkeler Date: Wed, 4 Mar 2015 21:21:30 -0500 Subject: Initial commit --- src/slalib/sun67.htx/node201.html | 215 ++++++++++++++++++++++++++++++++++++++ 1 file changed, 215 insertions(+) create mode 100644 src/slalib/sun67.htx/node201.html (limited to 'src/slalib/sun67.htx/node201.html') diff --git a/src/slalib/sun67.htx/node201.html b/src/slalib/sun67.htx/node201.html new file mode 100644 index 0000000..4613125 --- /dev/null +++ b/src/slalib/sun67.htx/node201.html @@ -0,0 +1,215 @@ + + + + +Using vectors + + + + + + + + + + + +

+ +

+
+ Next: Celestial Coordinate Systems +
+Up: Vectors and Matrices +
+ Previous: Vectors and Matrices +

+Using vectors +

+SLALIB provides conversions between spherical and vector +form +(sla_CS2C, +sla_CC2S +etc.), plus an assortment +of standard vector and matrix operations +(sla_VDV, +sla_MXV +etc.). +There are also routines +(sla_EULER +etc.) for creating a rotation matrix +from three Euler angles (successive rotations about +specified Cartesian axes). Instead of Euler angles, a rotation +matrix can be expressed as an axial vector (the pole of the rotation, +and the amount of rotation), and routines are provided for this +(sla_AV2M, +sla_M2AV +etc.). +

+Here is an example where spherical coordinates P1 and Q1 +undergo a coordinate transformation and become P2 and Q2; +the transformation consists of a rotation of the coordinate system +through angles A, B and C about the +z, new y and new z axes respectively: +

+            REAL A,B,C,R(3,3),P1,Q1,V1(3),V2(3),P2,Q2
+             :
+      *  Create rotation matrix
+            CALL sla_EULER('ZYZ',A,B,C,R)
+
+      *  Transform position (P,Q) from spherical to Cartesian
+            CALL sla_CS2C(P1,Q1,V1)
+
+      *  Multiply by rotation matrix
+            CALL sla_MXV(R,V1,V2)
+
+      *  Back to spherical
+            CALL sla_CC2S(V2,P2,Q2)
+

+Small adjustments to the direction of a position +vector are often most conveniently described in terms of + $[\,\Delta x,\Delta y, \Delta z\,]$ . Adding the correction +vector needs careful handling if the position +vector is to remain of length unity, an advisable precaution which +ensures that +the $[\,x,y,z\,]$ components are always available to mean the cosines of +the angles between the vector and the axis concerned. Two types +of shifts are commonly used, +the first where a small vector of arbitrary direction is +added to the unit vector, and the second where there is a displacement +in the latitude coordinate (declination, elevation etc.) alone. +

+For a shift produced by adding a small $[\,x,y,z\,]$ vector ${\bf D}$ to a +unit vector ${\bf V1}$ , the resulting vector ${\bf V2}$ has direction + $<{\bf V1}+{\bf D}\gt$ but is no longer of unit length. A better approximation +is available if the result is multiplied by a scaling factor of + $(1-{\bf D}\cdot{\bf V1})$ , where the dot +means scalar product. In Fortran: +

+            F = (1D0-(DX*V1X+DY*V1Y+DZ*V1Z))
+            V2X = F*(V1X+DX)
+            V2Y = F*(V1Y+DY)
+            V2Z = F*(V1Z+DZ)
+

+The correction for diurnal aberration (discussed later) is +an example of this form of shift. +

+As an example of the second kind of displacement +we will apply a small change in elevation $\delta E$ to an + $[\,Az,El~]$ direction vector. The direction of the +result can be obtained by making the allowable approximation + ${\tan \delta E\approx\delta E}$ and adding a adjustment +vector of length $\delta E$ normal +to the direction vector in the vertical plane containing the direction +vector. The z-component of the adjustment vector is + $\delta E \cos E$ ,and the horizontal component is + $\delta E \sin E$ which has then to be +resolved into x and y in proportion to their current sizes. To +approximate a unit vector more closely, a correction factor of + $\cos \delta E$ can then be applied, which is nearly + $(1-\delta E^2 /2)$ for +small $\delta E$ . Expressed in Fortran, for initial vector +V1X,V1Y,V1Z, change in elevation DEL +(+ve $\equiv$ upwards), and result +vector V2X,V2Y,V2Z: +

+            COSDEL = 1D0-DEL*DEL/2D0
+            R1 = SQRT(V1X*V1X+V1Y*V1Y)
+            F = COSDEL*(R1-DEL*V1Z)/R1
+            V2X = F*V1X
+            V2Y = F*V1Y
+            V2Z = COSDEL*(V1Z+DEL*R1)
+

+An example of this type of shift is the correction for atmospheric +refraction (see later). +Depending on the relationship between $\delta E$ and E, special +handling at the pole (the zenith for our example) may be required. +

+SLALIB includes routines for the case where both a position +and a velocity are involved. The routines +sla_CS2C6 +and +sla_CC62S +convert from $[\theta,\phi,\dot{\theta},\dot{\phi}]$ to $[\,x,y,z,\dot{x},\dot{y},\dot{z}\,]$ and back; +sla_DCS26 +and +sla_DC62S +are double precision equivalents. +

+ +

+
+ Next: Celestial Coordinate Systems +
+Up: Vectors and Matrices +
+ Previous: Vectors and Matrices +

+SLALIB --- Positional Astronomy Library
Starlink User Note 67
P. T. Wallace
12 October 1999
E-mail:ptw@star.rl.ac.uk +

+ + -- cgit