Fit an rectangle into another rectangle
Task is to fit a rectangle with given width 'w' into another rectangle with
given size 'a' x 'b', so that all four edges of inner rectangle touches the
outer rectangle ( at [x,0],[0,y],[a-x,b],[a,b-y] ). Now I need a formula to
calculate x and y from given a,b,w
http://forum.openscad.org/file/t2988/rect-in-rect.png
I have made an animation, which illustrates the problem.
(OpenSCAD script below)
If 'x' is known, solution is simple to calculate 'y' and 'w'. (see
fit_for_given_x(x) )
Now I need a function fit_for_given_w(w) to calculate 'x' and 'y'
--
Sent from: http://forum.openscad.org/
I think you need to know the height of the internal rectangle as there are
many ways to place one of width w that give different heights.
On Thu, 25 Mar 2021 at 09:27, bassklampfe jjvb-openscad@bassklampfe.de
wrote:
Task is to fit a rectangle with given width 'w' into another rectangle
with given size 'a' x 'b', so that all four edges of inner rectangle
touches the outer rectangle ( at [x,0],[0,y],[a-x,b],[a,b-y] ). Now I need
a formula to calculate x and y from given a,b,w
I have made an animation, which illustrates the problem.
(OpenSCAD script below)
If 'x' is known, solution is simple to calculate 'y' and 'w'. (see
fit_for_given_x(x) )
Now I need a function fit_for_given_w(w) to calculate 'x' and 'y'
// Animation: FPS=5 STEPS=100
a=13;
b=15;
module fit_for_given_x(x)
{
y=b-(b/2+sqrt(xx+bb/4-ax));
w=sqrt(xx+y*y);
inner_rect=[[x,0],[0,y],[a-x,b],[a,b-y]];
color("blue")linear_extrude(0.01)polygon(points=inner_rect);
color("black")translate([x,0,0])linear_extrude(0.02)text(str("x=",x),size=1,halign="center",valign="top");
color("black")translate([0,y,0])linear_extrude(0.02)text(str("y=",y),size=1,halign="right",valign="center");
color("black")translate([x/2,y/2,0])linear_extrude(0.02)text(str("w=",w),size=1,halign="left",valign="bottom");
}
if(is_num($t))
{
if($t>0)if($t<1)
{
fit_for_given_x(a*$t);
}
}
color("black")translate([a/2,b,0])linear_extrude(0.02)text(str("a=",a),size=1,halign="center",valign="bottom");
color("black")translate([a,b/2,0])linear_extrude(0.02)text(str("b=",b),size=1,halign="left",valign="center");
outer_rect=[[0,0],[0,b],[a,b],[a,0]];
color("yellow")linear_extrude(0.001)polygon(points=outer_rect);
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Nope. As long as 'a' < 'b' there is only one solution of 'x' and 'y' for a
given 'w'. Animation makes this pretty clear. (Please keep in mind, we talk
about rectangles)
Or can you show me a second solution for a=13,b=15,w=9.16399 (from
screenshot)? I'd be very interested.
If 'a' > 'b' there are some 'w' values, which have no possible no solution,
(in case of 'xx+bb/4-a*x' gets negative) but I don't care about these,
because I can assure, 'a' < 'b' for my special use case.
--
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Are you asserting that if the inner rectangle had half the height, all 4
corners would touch the outer edges in the same places? I am with nop head
on this, the ratio of w to h matters.
On Thu, 25 Mar 2021 at 10:18, bassklampfe jjvb-openscad@bassklampfe.de
wrote:
Nope. As long as 'a' < 'b' there is only one solution of 'x' and 'y' for a
given 'w'. Animation makes this pretty clear. (Please keep in mind, we talk
about rectangles)
Or can you show me a second solution for a=13,b=15,w=9.16399 (from
screenshot)? I'd be very interested.
If 'a' > 'b' there are some 'w' values, which have no possible no
solution, (in case of 'xx+bb/4-a*x' gets negative) but I don't care about
these, because I can assure, 'a' < 'b' for my special use case.
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Question: Have you LOOKED at the animation??? It explains more than 1000
words.
The both restrictions
- both outer and inner must be rectangular
- inner must touch outer with one edge at each side
enforce one only solution for x.
If I have x, I can easily calculate
y=b-(b/2+sqrt(x*(x-a)+b*b/4));
and I can also calculate the length of the other side of the inner
rectangle (let's call it v)
v=sqrt((a-x)(a-x)+(b-y)(b-y));
Am 25.03.21 um 11:43 schrieb Will Hardiman:
Are you asserting that if the inner rectangle had half the height, all
4 corners would touch the outer edges in the same places? I am with
nop head on this, the ratio of w to h matters.
Yes if you specify x and y there is only one w and h that works but if you
just have a line of length w you can place it at any x and get a
corresponding y and then project at right angles to get the other two
points giving different h values depending on the initial x.
On Thu, 25 Mar 2021 at 11:01, jjvb-openscad@bassklampfe.de wrote:
Question: Have you LOOKED at the animation??? It explains more than 1000
words.
The both restrictions
- both outer and inner must be rectangular
- inner must touch outer with one edge at each side
enforce one only solution for x.
If I have x, I can easily calculate
y=b-(b/2+sqrt(x*(x-a)+b*b/4));
and I can also calculate the length of the other side of the inner
rectangle (let's call it v)
v=sqrt((a-x)(a-x)+(b-y)(b-y));
Am 25.03.21 um 11:43 schrieb Will Hardiman:
Are you asserting that if the inner rectangle had half the height, all
4 corners would touch the outer edges in the same places? I am with
nop head on this, the ratio of w to h matters.
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Sorry, but no.
I did as you suggested. Lets have a=13, b=15, w=9 (to be similar to
screenshot of animation)
I place the line at an arbitrary location touching a and b, create two right
angles. (did this with Inkscape).
But – SURPRISE – they don't fit on the other side creating a rectangle.
See
http://forum.openscad.org/file/t2988/Other-W.png
There is only ONE x, where the two dotted lines will match, believe me...
--
Sent from: http://forum.openscad.org/
Ok, yes I see the problem now.
I think by symmetry and similar triangles y / x = (a-x) / (b- y). And x^2 +
y^2 = w^2, so you have to solve those two simultaneous equations.
On Thu, 25 Mar 2021 at 14:27, bassklampfe jjvb-openscad@bassklampfe.de
wrote:
Sorry, but no.
I did as you suggested. Lets have a=13, b=15, w=9 (to be similar to
screenshot of animation)
I place the line at an arbitrary location touching a and b, create two
right angles. (did this with Inkscape).
But – SURPRISE – they don't fit on the other side creating a rectangle.
See
There is only ONE x, where the two dotted lines will match, believe me...
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I tried to solve this algebraically, and ended up with a quartic in x that
doesn't look like it has easy solutions at first glance:
4x^4 - 4a x^3 + (a^2 + b^2 - 4w^2 ) x^2 + 2 w a x + (w^4- b^2 w^2) = 0
--
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When you've solved that, then consider a room, say 14ft by 12ft, and you
need to cut a scaffold plank, 1 foot wide, with square ends, so that it
fits exactly diagonally across the corners, as a platform for artexing
the ceiling, say. What length plank is required? (The four corners of
the plank touch the walls). Has to be a formula, no iterative
process/drawing allowed.
On 25/03/2021 09:26, bassklampfe wrote:
Task is to fit a rectangle with given width 'w' into another rectangle
with given size 'a' x 'b', so that all four edges of inner rectangle
touches the outer rectangle ( at [x,0],[0,y],[a-x,b],[a,b-y] ). Now I
need a formula to calculate x and y from given a,b,w
I have made an animation, which illustrates the problem.
(OpenSCAD script below)
If 'x' is known, solution is simple to calculate 'y' and 'w'. (see
fit_for_given_x(x) )
Now I need a function fit_for_given_w(w) to calculate 'x' and 'y'
// Animation: FPS=5 STEPS=100
a=13;
b=15;
module fit_for_given_x(x)
{
y=b-(b/2+sqrt(xx+bb/4-ax));
w=sqrt(xx+y*y);
inner_rect=[[x,0],[0,y],[a-x,b],[a,b-y]];
color("blue")linear_extrude(0.01)polygon(points=inner_rect);
color("black")translate([x,0,0])linear_extrude(0.02)text(str("x=",x),size=1,halign="center",valign="top");
color("black")translate([0,y,0])linear_extrude(0.02)text(str("y=",y),size=1,halign="right",valign="center");
color("black")translate([x/2,y/2,0])linear_extrude(0.02)text(str("w=",w),size=1,halign="left",valign="bottom");
}
if(is_num($t))
{
if($t>0)if($t<1)
{
fit_for_given_x(a*$t);
}
}
color("black")translate([a/2,b,0])linear_extrude(0.02)text(str("a=",a),size=1,halign="center",valign="bottom");
color("black")translate([a,b/2,0])linear_extrude(0.02)text(str("b=",b),size=1,halign="left",valign="center");
outer_rect=[[0,0],[0,b],[a,b],[a,0]];
color("yellow")linear_extrude(0.001)polygon(points=outer_rect);
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I gave the equation to Wolfram to solve. It did but answer is horrible:
https://www.wolframalpha.com/input/?i=solve+sqrt%28w%5E2+-x%5E2%29%2Fx+%3D+%28a-x%29+%2F+%28b+-+sqrt%28w%5E2+-x%5E2%29%29
On Thu, 25 Mar 2021 at 17:08, Ray West raywest@raywest.com wrote:
When you've solved that, then consider a room, say 14ft by 12ft, and you
need to cut a scaffold plank, 1 foot wide, with square ends, so that it
fits exactly diagonally across the corners, as a platform for artexing the
ceiling, say. What length plank is required? (The four corners of the
plank touch the walls). Has to be a formula, no iterative process/drawing
allowed.
On 25/03/2021 09:26, bassklampfe wrote:
Task is to fit a rectangle with given width 'w' into another rectangle
with given size 'a' x 'b', so that all four edges of inner rectangle
touches the outer rectangle ( at [x,0],[0,y],[a-x,b],[a,b-y] ). Now I need
a formula to calculate x and y from given a,b,w
I have made an animation, which illustrates the problem.
(OpenSCAD script below)
If 'x' is known, solution is simple to calculate 'y' and 'w'. (see
fit_for_given_x(x) )
Now I need a function fit_for_given_w(w) to calculate 'x' and 'y'
// Animation: FPS=5 STEPS=100
a=13;
b=15;
module fit_for_given_x(x)
{
y=b-(b/2+sqrt(xx+bb/4-ax));
w=sqrt(xx+y*y);
inner_rect=[[x,0],[0,y],[a-x,b],[a,b-y]];
color("blue")linear_extrude(0.01)polygon(points=inner_rect);
color("black")translate([x,0,0])linear_extrude(0.02)text(str("x=",x),size=1,halign="center",valign="top");
color("black")translate([0,y,0])linear_extrude(0.02)text(str("y=",y),size=1,halign="right",valign="center");
color("black")translate([x/2,y/2,0])linear_extrude(0.02)text(str("w=",w),size=1,halign="left",valign="bottom");
}
if(is_num($t))
{
if($t>0)if($t<1)
{
fit_for_given_x(a*$t);
}
}
color("black")translate([a/2,b,0])linear_extrude(0.02)text(str("a=",a),size=1,halign="center",valign="bottom");
color("black")translate([a,b/2,0])linear_extrude(0.02)text(str("b=",b),size=1,halign="left",valign="center");
outer_rect=[[0,0],[0,b],[a,b],[a,0]];
color("yellow")linear_extrude(0.001)polygon(points=outer_rect);
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No offense, but this whole thing starts feeling like we are doing
somebody's homework here.
Regards, Jan
On 3/25/21 1:08 PM, nop head wrote:
I gave the equation to Wolfram to solve. It did but answer is horrible:
https://www.wolframalpha.com/input/?i=solve+sqrt%28w%5E2+-x%5E2%29%2Fx+%3D+%28a-x%29+%2F+%28b+-+sqrt%28w%5E2+-x%5E2%29%29
https://www.wolframalpha.com/input/?i=solve+sqrt%28w%5E2+-x%5E2%29%2Fx+%3D+%28a-x%29+%2F+%28b+-+sqrt%28w%5E2+-x%5E2%29%29
On Thu, 25 Mar 2021 at 17:08, Ray West <raywest@raywest.com
mailto:raywest@raywest.com> wrote:
When you've solved that, then consider a room, say 14ft by 12ft, and
you need to cut a scaffold plank, 1 foot wide, with square ends, so
that it fits exactly diagonally across the corners, as a platform
for artexing the ceiling, say. What length plank is required? (The
four corners of the plank touch the walls). Has to be a formula, no
iterative process/drawing allowed.
On 25/03/2021 09:26, bassklampfe wrote:
Task is to fit a rectangle with given width 'w' into another
rectangle with given size 'a' x 'b', so that all four edges of
inner rectangle touches the outer rectangle ( at
[x,0],[0,y],[a-x,b],[a,b-y] ). Now I need a formula to calculate x
and y from given a,b,w
I have made an animation, which illustrates the problem.
(OpenSCAD script below)
If 'x' is known, solution is simple to calculate 'y' and 'w'. (see
fit_for_given_x(x) )
Now I need a function fit_for_given_w(w) to calculate 'x' and 'y'
// Animation: FPS=5 STEPS=100
a=13;
b=15;
module fit_for_given_x(x)
{
y=b-(b/2+sqrt(x*x+b*b/4-a*x));
w=sqrt(x*x+y*y);
inner_rect=[[x,0],[0,y],[a-x,b],[a,b-y]];
color("blue")linear_extrude(0.01)polygon(points=inner_rect);
color("black")translate([x,0,0])linear_extrude(0.02)text(str("x=",x),size=1,halign="center",valign="top");
color("black")translate([0,y,0])linear_extrude(0.02)text(str("y=",y),size=1,halign="right",valign="center");
color("black")translate([x/2,y/2,0])linear_extrude(0.02)text(str("w=",w),size=1,halign="left",valign="bottom");
}
if(is_num($t))
{
if($t>0)if($t<1)
{
fit_for_given_x(a*$t);
}
}
color("black")translate([a/2,b,0])linear_extrude(0.02)text(str("a=",a),size=1,halign="center",valign="bottom");
color("black")translate([a,b/2,0])linear_extrude(0.02)text(str("b=",b),size=1,halign="left",valign="center");
outer_rect=[[0,0],[0,b],[a,b],[a,0]];
color("yellow")linear_extrude(0.001)polygon(points=outer_rect);
------------------------------------------------------------------------
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--
Jan Wieck
Principle Database Engineer
Amazon Web Services
I stopped and threw away my work when I saw that you had gotten to the
same "two equations and two unknowns" that I started with. I didn't yet
have a simple expression, but where I stopped I had fairly
straightforward case of the quadratic function. I was pretty sure it
was uglier than it needed to be, but didn't feel like spending more time
simplifying - and it seemed likely that somebody else would do it easier
and faster. It was maybe half a line of text long at that point. I'm
pretty confident that the math salad that Wolfram came up with is
unnecessary.
nophead wrote
I gave the equation to Wolfram to solve. It did but answer is horrible:
https://www.wolframalpha.com/input/?i=solve+sqrt%28w%5E2+-x%5E2%29%2Fx+%3D+%28a-x%29+%2F+%28b+-+sqrt%28w%5E2+-x%5E2%29%29
Your input seems better input as what I got for
(w² − x²)^(0.5) = − ( − b + (b² + 4(x² − ax))^(0.5))/2
at https://www.dcode.fr/equation-solver
I will try, if the solution from wolframalpha is correct thanks
--
Sent from: http://forum.openscad.org/
MostlyHarmless wrote
No offense, but this whole thing starts feeling like we are doing
somebody's homework here.
I can tell you, you are not.
Solution of this problem is one missing part to build steel constructions
like
http://forum.openscad.org/file/t2988/peine-profiles-example.png
without mechanical errors and in the final step to build full customizable
railway steel bridges in OpenSCAD.
When all things are completed, I will happily share the result with you.
--
Sent from: http://forum.openscad.org/
nophead wrote
I gave the equation to Wolfram to solve. It did but answer is horrible:
https://www.wolframalpha.com/input/?i=solve+sqrt%28w%5E2+-x%5E2%29%2Fx+%3D+%28a-x%29+%2F+%28b+-+sqrt%28w%5E2+-x%5E2%29%29
I'm not scared by the answer, as long as it is correct – which still has to
be prooved.
But isn't it this, why we use computers? To do calculations, which feels
like horror to some people
--
Sent from: http://forum.openscad.org/
On 3/25/21 2:28 PM, bassklampfe wrote:
MostlyHarmless wrote
No offense, but this whole thing starts feeling like we are doing
somebody's homework here.
I can tell you, you are not.
Solution of this problem is one missing part to build steel
constructions like
without mechanical errors and in the final step to build full
customizable railway steel bridges in OpenSCAD.
As said, no offense. I'm just around the interwebs since the late 80's
(yeah, it was email and netnews over UUCP back then), so I have seen
this sort of math problem without explaining a practical purpose once or
twice before ;)
My gut feeling is that it requires taking the angles into account. The
inner rectangle has a certain rotation that produces 4 right triangles,
which all have identical angles. I'll have to think about it a bit
longer, but maybe that will give us the 3rd equation to eliminate one of
the variables.
Regards, Jan
When all things are completed, I will happily share the result with you.
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--
Jan Wieck
Principle Database Engineer
Amazon Web Services
Nop head was on the right path. But he left the a,b,w as variables for
Wolfram. It doesn't complete within the allotted time, so fails to
simplify. I tried specifying the example input variables of A=13, B=15
and W=9.16, which allows the algorithm to simplify earlier. Now it solves
for x=8.57 (and also some imaginary solutions).
The original request was for a formula to set up Fn(a,b,w) returning x. I
am not able to solve the problem the way Wolfram did. It appears to use 2
simultaneous equations (see the two lines on the graph). Nop head
specified those.
Interesting problem that seems like it out to be easy, but seem to actually
be computationally complex.
Kevin
On Thu, Mar 25, 2021 at 1:10 PM nop head nop.head@gmail.com wrote:
I gave the equation to Wolfram to solve. It did but answer is horrible:
https://www.wolframalpha.com/input/?i=solve+sqrt%28w%5E2+-x%5E2%29%2Fx+%3D+%28a-x%29+%2F+%28b+-+sqrt%28w%5E2+-x%5E2%29%29
On Thu, 25 Mar 2021 at 17:08, Ray West raywest@raywest.com wrote:
When you've solved that, then consider a room, say 14ft by 12ft, and you
need to cut a scaffold plank, 1 foot wide, with square ends, so that it
fits exactly diagonally across the corners, as a platform for artexing the
ceiling, say. What length plank is required? (The four corners of the
plank touch the walls). Has to be a formula, no iterative process/drawing
allowed.
On 25/03/2021 09:26, bassklampfe wrote:
Task is to fit a rectangle with given width 'w' into another rectangle
with given size 'a' x 'b', so that all four edges of inner rectangle
touches the outer rectangle ( at [x,0],[0,y],[a-x,b],[a,b-y] ). Now I need
a formula to calculate x and y from given a,b,w
I have made an animation, which illustrates the problem.
(OpenSCAD script below)
If 'x' is known, solution is simple to calculate 'y' and 'w'. (see
fit_for_given_x(x) )
Now I need a function fit_for_given_w(w) to calculate 'x' and 'y'
// Animation: FPS=5 STEPS=100
a=13;
b=15;
module fit_for_given_x(x)
{
y=b-(b/2+sqrt(xx+bb/4-ax));
w=sqrt(xx+y*y);
inner_rect=[[x,0],[0,y],[a-x,b],[a,b-y]];
color("blue")linear_extrude(0.01)polygon(points=inner_rect);
color("black")translate([x,0,0])linear_extrude(0.02)text(str("x=",x),size=1,halign="center",valign="top");
color("black")translate([0,y,0])linear_extrude(0.02)text(str("y=",y),size=1,halign="right",valign="center");
color("black")translate([x/2,y/2,0])linear_extrude(0.02)text(str("w=",w),size=1,halign="left",valign="bottom");
}
if(is_num($t))
{
if($t>0)if($t<1)
{
fit_for_given_x(a*$t);
}
}
color("black")translate([a/2,b,0])linear_extrude(0.02)text(str("a=",a),size=1,halign="center",valign="bottom");
color("black")translate([a,b/2,0])linear_extrude(0.02)text(str("b=",b),size=1,halign="left",valign="center");
outer_rect=[[0,0],[0,b],[a,b],[a,0]];
color("yellow")linear_extrude(0.001)polygon(points=outer_rect);
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Maybe there's a clever way to solve this using trig, but I'm not seeing
it---and the similarity relationship really seems like the fundamental
defining property.
Solving a quartic isn't a big deal. But solving one symbolically like
Wolfram Alpha does is completely pointless. The quartic formula is a
horrible mess. Give up on the idea of having a formula, and definitely
don't try to use the mess from Worlfram Alpha. You need to solve the
equation numerically. BOSL2 has a polynomial root finder that will do the
job.
https://github.com/revarbat/BOSL2/wiki/math.scad#function-real_roots
include<BOSL2/std.scad>
a=13;
b=15;
w=9;
function getx(a,b,w) =
let(
options = real_roots([4,-4a, aa+bb-4ww, 2wwa,
pow(w,4)-bbw*w]),
posopt = [for (x=options) if (x>0) x]
)
assert(len(posopt)==1)
posopt[0];
x=getx(a,b,w);
y=b-(b/2+sqrt(xx+bb/4-a*x));
echo(x=x,y=y);
echo(sqrt(sqr(x)+sqr(y)));
inner = [[x,0], [0, y], [a-x,b], [a,b-y]];
linear_extrude(height=1)square([a,b]);
color("red")linear_extrude(height=1.1)polygon(inner);
http://forum.openscad.org/file/t2477/rect.png
kdtop wrote
Nop head was on the right path. But he left the a,b,w as variables for
Wolfram. It doesn't complete within the allotted time, so fails to
simplify. I tried specifying the example input variables of A=13, B=15
and W=9.16, which allows the algorithm to simplify earlier. Now it solves
for x=8.57 (and also some imaginary solutions).
The original request was for a formula to set up Fn(a,b,w) returning x. I
am not able to solve the problem the way Wolfram did. It appears to use 2
simultaneous equations (see the two lines on the graph). Nop head
specified those.
Interesting problem that seems like it out to be easy, but seem to
actually
be computationally complex.
Kevin
On Thu, Mar 25, 2021 at 1:10 PM nop head <
nop.head@
> wrote:
I gave the equation to Wolfram to solve. It did but answer is horrible:
https://www.wolframalpha.com/input/?i=solve+sqrt%28w%5E2+-x%5E2%29%2Fx+%3D+%28a-x%29+%2F+%28b+-+sqrt%28w%5E2+-x%5E2%29%29
On Thu, 25 Mar 2021 at 17:08, Ray West <
raywest@
> wrote:
When you've solved that, then consider a room, say 14ft by 12ft, and you
need to cut a scaffold plank, 1 foot wide, with square ends, so that it
fits exactly diagonally across the corners, as a platform for artexing
the
ceiling, say. What length plank is required? (The four corners of the
plank touch the walls). Has to be a formula, no iterative
process/drawing
allowed.
On 25/03/2021 09:26, bassklampfe wrote:
Task is to fit a rectangle with given width 'w' into another rectangle
with given size 'a' x 'b', so that all four edges of inner rectangle
touches the outer rectangle ( at [x,0],[0,y],[a-x,b],[a,b-y] ). Now I
need
a formula to calculate x and y from given a,b,w
I have made an animation, which illustrates the problem.
(OpenSCAD script below)
If 'x' is known, solution is simple to calculate 'y' and 'w'. (see
fit_for_given_x(x) )
Now I need a function fit_for_given_w(w) to calculate 'x' and 'y'
// Animation: FPS=5 STEPS=100
a=13;
b=15;
module fit_for_given_x(x)
{
y=b-(b/2+sqrt(xx+bb/4-ax));
w=sqrt(xx+y*y);
inner_rect=[[x,0],[0,y],[a-x,b],[a,b-y]];
color("blue")linear_extrude(0.01)polygon(points=inner_rect);
color("black")translate([x,0,0])linear_extrude(0.02)text(str("x=",x),size=1,halign="center",valign="top");
color("black")translate([0,y,0])linear_extrude(0.02)text(str("y=",y),size=1,halign="right",valign="center");
color("black")translate([x/2,y/2,0])linear_extrude(0.02)text(str("w=",w),size=1,halign="left",valign="bottom");
}
if(is_num($t))
{
if($t>0)if($t<1)
{
fit_for_given_x(a*$t);
}
}
color("black")translate([a/2,b,0])linear_extrude(0.02)text(str("a=",a),size=1,halign="center",valign="bottom");
color("black")translate([a,b/2,0])linear_extrude(0.02)text(str("b=",b),size=1,halign="left",valign="center");
outer_rect=[[0,0],[0,b],[a,b],[a,0]];
color("yellow")linear_extrude(0.001)polygon(points=outer_rect);
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nophead wrote
I gave the equation to Wolfram to solve. It did but answer is horrible:
https://www.wolframalpha.com/input/?i=solve+sqrt%28w%5E2+-x%5E2%29%2Fx+%3D+%28a-x%29+%2F+%28b+-+sqrt%28w%5E2+-x%5E2%29%29
When you described it as "horrible", I expected something more along the
lines of "a bit awkward", but horrible does not give that solution justice.
This goes beyond horrible, and into pure terror...
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