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.


geometry_2021Jun16a.gh (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.


geometry_2021Jun16b.gh (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

study.gh (24.8 KB)

Check this

find top point.gh (24.8 KB)

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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
H
c = N
M/2
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

image

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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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