I have a hexagon that sits on a side (as opposed to sitting on a
vertice - verticies are at the 1, 3,5, 7, 9, and 11 o?clock
positions). The topside is length 's' the bottom side is length 'c'.
's' does not equal 'c'. The 'hips' of the hexagon have two sides that
face 'up' of length 'a' and two that face down of length 'b'. 'a'
does not equal 'b'. The sides in clockwise order (from the top) are
s-a-b-c-b-a.
The overall height of the hexagon is assumed to be 'Y'. The center
points of topside 's' and bottom 'c' are perpendicular to each other,
making the polygon symmetric about its vertical axis. I have labeled
the vertices on the right hand side i, j, k clockwise; on the left
hand side i?, j?, k? counter-clockwise.
I am trying to formulate a set of equations or a single function that
relates Y, a, b, c, and s with no additional variables. Geometrically
I bisected the hex down the middle vertically along the center points
of 's' and 'c' and horizontally from the vertice of the 'left hip' to
that of the 'right hip'. I formed interior triangles and used a^2 +
b^2 - 2abcos(A) = c^2 to form relationships between the sides.
I came up with several equations using new variables to describe the
interior triangles. I introduced angles A and B as the angles between
the horizontal bisector and the upward facing side i-j (length 'a')
and the downward facing side j-k (length 'b'), respectively. I got 3
unknowns in three equations, but have not been able to solve for an
equation with only the variables a, b, c, s, and Y. These are the
equations I derived, but still I cannot simplify:
(Y-a*sin(A))^2 + b^2*cos^2(B) = b^2
b*sinB + a*sinA = Y
Y^2 + ((c-s)/2)^2 = a^2 + b^2 ? 2*a*b*cos(A+B)
(c-s)/2 = a/(cosA) ? b/(cosB)
I even introduced another variable X, the width of the hexagon from j
to j? getting the relationships
(X/2 ? c/2)*cos B = b
(X/2 ? s/2)*cos A = a
Any takers? |
Request for Question Clarification by
mathtalk-ga
on
15 Feb 2004 07:55 PST
Hi, slakeu-ga:
In describing the vertex positions as being in "clock positions", are
you just being suggestive, or do you mean to imply that the hexagon is
inscribed in a circle?
Not every hexagon, nor even every bilaterally symmetric one (as yours
is), can be inscribed in a circle. So requiring the vertices to lie
on a circle would reduce the number of total unknowns needed to
describe your figure, or to look at it another way, it will increase
the number of formulas that can be deduced for the coordinates of the
vertices.
regards, mathtalk-ga
|
Request for Question Clarification by
mathtalk-ga
on
16 Feb 2004 15:48 PST
Hi, slakeu-ga:
I've read through your description pretty thoroughly and feel that
I've got a good idea of what you'd ideally like to do, for example to
express the total height Y of the hexagon as a function of side
lengths a,b,c, and s.
However it is not possible to do so without additional information of
some sort. If one "grabs" the hexagon by the "hips" at j,j', then it
can be "played" like an accordian, alternating increasing or
decreasing the height as the points j,j' move in or out, respectively.
At best one can give feasible limits (inequalities) on Y which bound
it above or below with functions of a,b,c, and s.
Hence the Request for Clarification that I posted earlier, asking
whether the vertices are constrained to lie on a circle. Such an
assumption would lead to a formula linking Y with a,b,c, and s that at
least implicitly determines Y as a function of the side lengths
(subject to some rather mild consistency conditions on a,b,c, and s).
regards, mathtalk-ga
|
