You are a field officer in the Greek artillery corps.

geometry/construction/artillery New

New entry, edited from material collected for the archive and not yet in a released version.

Problem

You are a field officer in the Greek artillery corps.  As shown in the
crude ascii sketch below, you set your cannon to an arbitrary elevation
(angle) and fire.  The shot misses the target T on an opposing hill and
hits the M shown on a more distant hill.  Given a scale side view of the
terrain (as below), showing the locations of the cannon, initial miss,
and target and a line corresponding to the initial angle of the cannon,
construct the two angles which will hit the target.  Naturally you can
use only a straight edge and compass.  Ignore air resistance and other
weird effects.

                                             ---M----
      E                                     /        \
     /                   --T---            /          \
    /                   /      \          /            \
---C-------L------------        ----------              -----------

-- Joe Riel <jsr@TorreyPinesCA.ncr.com>

Show solution

Given: Point C (location of cannon), points M and T, line CL
(horizontal), line CE (tangent to trajectory that hits M).

Extend a perpendicular to line CL upward through M so that it intersects
line CE at point A (directly above M) and line CL at point B (directly
below M).

Draw the perpendicular bisector of segment CA and find its
intersection with line MA.  Call this point D.  Now place point G on
the line between C and D such that CG = MB.  Extend the perpendicular
bisector of segment GM until it intersects line CD at F.  Point F is
the focus of a parabola passing through C, tangent to CE, and through
M.  (In fact, this parabola is the path of the first shot.)

Mark point N on the line BA above B such that BN = 2(CF).  Also, draw
a line through T and M.  Now draw a line perpendicular to line BN
through N (this is a horizontal line), and draw another line through C
perpendicular to line MT.  Extend these two newest lines until they
intersect, and call this point O (this is a point on the line through
C perpendicular to line MT, at a height 2(CF) above the ground).
Draw a circle with center O and radius OA.

Extend a perpendicular to line CL upward through T until it intersects
the circle about O at points X and Y.

Aim the cannon at either point X or Y in order to hit point T.

This construction obviously must fail if point T is outside the range
of the cannon.  It also fails if point T is directly above or below M,
that is, if line MT is perpendicular to line CL.

There is yet another alternative construction that works even if M and
T are on the same vertical line.  Follow the previous construction
until you find point F.  Then drop a perpendicular to line CL through
point T, intersecting CL at point S.  Place Y between C and F such
that CY = ST.  Draw a circle about T with radius YF and another about
C with radius CF.  The two circles intersect at points V and W (which
may be the same point).  Erect a perpendicular to line CL through C,
and label a point U on this line above C.

Aim that cannon along either the bisector of angle UCV or the bisector
of angle UCW to hit point T.

Again, I leave it as an exercise to show the constructions correct or
incorrect.

-- David A. Karr (karr@cs.cornell.edu)