Geometry problem

Hi all!

I’m currently stuck with a geometric construction like this:

H, T and A (the area of the black shape) are known.
I need to define the shape, finding the position of the top vertex.
If A is larger than the area of the H-T triangle (H*T/2), there should be only one solution, but how to find it?
I’m searching for a math function to solve this… possibly non-iterative.

(This is actually the derivative graph of an animation for a web page :sweat_smile: , but that is unrelated now.)

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Here is a model that accepts T, H and ‘Half Angle’ to compute Area. (11.8 KB)

Maybe what you want will occur to me after coffee. :wink:

The Area is given, not the angle :slight_smile:

I know that!

I never drink coffee… I probably drank it like 2 times in my life.
When i need to “wake up” (or i’m bored) i search for pseudo-easy tasks on this forum… brain fuel!

I hope that magic drink does the trick for you!

The obvious answer is to use Range to get a series of half angles, compute Area for all of them, sort to find closest area and corresponding half angle.

k is known too…

that is sort of iterative… and this function will be executed by javascript at 60fps on browser.
I need it to be a one-shot with fewer math possible.

Oh well. (20.4 KB)

Slider in yellow group is ‘TARGET AREA’. ‘Steps’ slider is a form of precision, which is also affected by the range bounds (‘Half Angle Min’ and ‘Half Angle Max’).

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2021-06-16 16_27_23-telaio fisso 009.3dm (79 MB) - Rhino 7 Commerciale - Frontale
with your script is visible that the path of the top vertex is a curve that is tangent to the mirror of the hypotenuse and goes up asymptotic to the vertical line starting from mid base… how can i convert to math this :crazy_face:

It also makes it obvious that things go goofy when the Half Angle goes above 80 in this example.

This solution using Galapagos (24.8 KB)

Check this

find top (24.8 KB)


I have had some gin tonics tonight, so this might very well be completely wrong :slight_smile:

we have the total area of the shape, that I interpret to be these three areas summed up together
we just know the area of 1, being H*T/2

and I think we might know all the angles here:

so if we write the area of figure 3 and figure 2 exclusively in relation of the edge P, we should get an a P^2 + b P + c equation that should be solvable… probably I should just go to sleep

once you have length of edge P, angle d is known, you can calculate the position of the tip of the shape
I have the terrible feeling tomorrow morning I’m gonna be very ashamed of this message

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I don’t use angles or complex math.
The shape divided to different areas and i create another area equation S which is a triangle.
a,b,c replaced by their equations.
a = (T-(2N))H/2
b = N
c = N
A (is already known) and A = a+2b+2c
S = a+1b+4c = T*(H+2M)/2
A (also) = S - 2c so: A = (T*(H+2M)/2) - 2c

Until we find the final equation which can solved easily.


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If the two angles at the top are the same, there are two other angles which are also equal. At the right end and left end above H. Triangles a and c are similar.

M / N = H / (T - 2N)

I don’t understand what you mean , a and c are not similar but they are in one case when H = M

The marked angles are all the same, therefore a is similar to c


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No, this condition is not true

I wrote similar, not equal.

When H = M, the two triangles a and c are equal or congruent.

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