Revision: 52785
http://brlcad.svn.sourceforge.net/brlcad/?rev=52785&view=rev
Author: n_reed
Date: 2012-10-04 19:23:40 +0000 (Thu, 04 Oct 2012)
Log Message:
-----------
explain some of the math
Modified Paths:
--------------
brlcad/trunk/src/librt/primitives/epa/epa.c
Modified: brlcad/trunk/src/librt/primitives/epa/epa.c
===================================================================
--- brlcad/trunk/src/librt/primitives/epa/epa.c 2012-10-04 17:43:08 UTC (rev
52784)
+++ brlcad/trunk/src/librt/primitives/epa/epa.c 2012-10-04 19:23:40 UTC (rev
52785)
@@ -772,6 +772,27 @@
return num_new_points;
}
+/* A canonical parabola in the Y-Z plane has equation z = y^2 / 4p, and opens
+ * toward positive z with vertex at the origin.
+ *
+ * The countour of an epa in the plane H-R (where R is one of the epa axes A or
+ * B) is a parabola with vertex at H, opening toward -H. We can transform this
+ * parabola to get an equivalent canonical parabola in the Y-Z plane, opening
+ * toward positive Z (-H) with vertex at the origin (H).
+ *
+ * This parabola passes through the point (r, |H|) (where r = |A| or |B|). If
+ * we plug the point (r, |H|) into our canonical equation, we see how p relates
+ * to r and |H|:
+ *
+ * |H| = r^2 / 4p
+ * p = (r^2) / (4|H|)
+ */
+static fastf_t
+epa_parabola_p(fastf_t r, fastf_t mag_h)
+{
+ return (r * r) / (4 * mag_h);
+}
+
static struct rt_pt_node *
epa_parabolic_curve(fastf_t mag_h, fastf_t r, int num_points)
{
@@ -785,7 +806,7 @@
VSET(curve->p, 0, 0, -mag_h);
VSET(curve->next->p, 0, r, 0);
- count = approximate_parabolic_curve(curve, (r * r) / (4 * mag_h),
num_points - 2);
+ count = approximate_parabolic_curve(curve, epa_parabola_p(r, mag_h),
num_points - 2);
if (count != (num_points - 2)) {
return NULL;
@@ -794,6 +815,29 @@
return curve;
}
+/* The countour of an epa in the plane H-R (where R is one of the epa axes A or
+ * B) is a parabola with vertex at H, opening toward -H. We can transform this
+ * parabola into an equivalent one in the Y-Z plane which has vertext at (0,
|H|)
+ * and opens toward -Z.
+ *
+ * The equation for this parabola is a variant of the equation for a canonical
+ * parabola in the Y-Z plane (z = y^2 / 4p):
+ * z = |H| - y^2 / 4p
+ *
+ * Solving this equation for y yields:
+ * y = sqrt(4p * (|H| - z))
+ *
+ * Substituting p = r^2 / 4|H| (see above comment):
+ * y = sqrt(r^2 * (|H| - z) / |H|)
+ * = r * sqrt((|H| - z) / |H|)
+ * = r * sqrt(1 - z / |H|)
+ */
+static fastf_t
+epa_parabola_y(fastf_t r, fastf_t mag_H, fastf_t z)
+{
+ return r * sqrt(1 - z / mag_H);
+}
+
/* Plot the elliptical cross section of the given epa at distance h along the
* epa height vector (h >= 0, h <= |H|) consisting of num_points points.
*/
@@ -816,8 +860,8 @@
/* calculate semi-major and semi-minor axis for the elliptical
* cross-section at distance h along H
*/
- VSCALE(A, Au, epa->epa_r1 * sqrt(1 - h / mag_H));
- VSCALE(B, Bu, epa->epa_r2 * sqrt(1 - h / mag_H));
+ VSCALE(A, Au, epa_parabola_y(epa->epa_r1, mag_H, h));
+ VSCALE(B, Bu, epa_parabola_y(epa->epa_r2, mag_H, h));
VJOIN1(cross_section_plane, V, h, Hu);
radian_step = bn_twopi / num_points;
@@ -852,13 +896,13 @@
VSCALE(Hu, epa->epa_H, 1.0 / mag_H);
z = pts->p[Z];
- VJOIN2(p, epa_V, r * sqrt(z / mag_H + 1), Ru, -z, Hu);
+ VJOIN2(p, epa_V, epa_parabola_y(r, mag_H, -z), Ru, -z, Hu);
RT_ADD_VLIST(vhead, p, BN_VLIST_LINE_MOVE);
node = pts->next;
while (node != NULL) {
z = node->p[Z];
- VJOIN2(p, epa_V, r * sqrt(z / mag_H + 1), Ru, -z, Hu);
+ VJOIN2(p, epa_V, epa_parabola_y(r, mag_H, -z), Ru, -z, Hu);
RT_ADD_VLIST(vhead, p, BN_VLIST_LINE_DRAW);
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