Difficult 3d calculation needed

[tmdarkmatter]

Hello,

I am currently dealing with a mathematical problem I am still not able to resolve:

Imagine there is a sphere with radius 1 and the center is just in the middle of the coordinate system, so the equation is (x - 0)^2 + (y - 0)^2 + (z - 0)^2 = r^2

And there is a point with the coordinates (5, 0, 0) which is the tip of a cone that is directed directly towards the center of the sphere till the end of the sphere.

What I now want to calculate is the exact angle this cone needs so that the volume it occupies within the sphere is exactly half of the sphere. And I need to figure out how long the line between the first intersection of the cone when hitting the sphere for the first time and the end of the cone on the other side of the sphere (second intersection) is.

Here I have a little drawing to better explain what I need:

The problem with this is that the remaining part of the sphere that is not within the cone is actually a ring going around the cone, making the calculation much more difficult.

I know that the further away the point P is from the center, the closer the distance between both intersections gets to exactly 2r. And this distance decreases very little to values from 2 to 1,6 when the point P gets closer to the sphere.

Another problem is that the area where the cone is touching the sphere is not even flat, so this would even include the calculation of two domes on both sides.

Please, any help will be highly appreciated!

 

[Genady]

I don't think the remaining part of the sphere matters at all. You need to express the volume of the part cut out by the cone and make it equal 2/3 pi r^3.

[tmdarkmatter]

Yes, that makes sense. But how do I express it when the ground and roof of that cone is a dome? And the angle of that cone still depends on the position of point P. At first I thought that this would be easy using the tan functions, considering that the cone would always hit the y axis at the same height, but not even that is the case. Even the height at the y axis goes up very little as point P gets closer to the center of the sphere.

Edited Friday at 10:38 PM by tmdarkmatter

[KJW]

https://en.wikipedia.org/wiki/Solid_of_revolution

 

[tmdarkmatter]

Yes, I will have to study this during the weekend. How I can apply this to my problem. Thank you! I hope I can solve it (or at least get close to a solution).

[KJW]

Deleted.

Edited Friday at 11:29 PM by KJW
Deleted due to error

[Genady]

56 minutes ago, tmdarkmatter said:

the ground and roof of that cone is a dome

Calculate them separately, e.g., The spherical dome: Surface area and volume

[tmdarkmatter]

Yes, it should be something like the double integral of the unknown equation minus the double integral of the sphere between -r and the second intersection minus the double integral of the sphere between the first intersection and +r is equal to 2/3 pi r^3, where the equation is -(b/d)x + b and actually what I am looking for of the equation is b which is the value of the equation at x0. d is the distance between x0 and the point P.

[Genady]

Do you need the integrals? The volume of the cone between the two cuts plus the volumes of the two domes equal 2/3 pi r^3. The formulas for all these volumes are known.

[Sensei]

Do you need a formula or an approximate value (calculated by a computer program) for a real-world application?

There are many tricks to write a program without creating mathematical formulas with enough precision..

 

I solve “unsolvable problems” with ease. A common example is dividing by zero. Any division by a variable must be checked before executing the code, otherwise an exception is raised (program crash and shutdown will occur).

So basically what is the purpose of this?

Can you write computer programs?

[tmdarkmatter]

32 minutes ago, Genady said:

Do you need the integrals? The volume of the cone between the two cuts plus the volumes of the two domes equal 2/3 pi r^3. The formulas for all these volumes are known.

The problem is that all of these volumes still have an unknown value for "b" that still has to be calculated. This b does not only modify the slope of the negative equation, it also moves the points of the intersections on the x axis.

22 minutes ago, Sensei said:

Do you need a formula or an approximate value (calculated by a computer program) for a real-world application?

There are many tricks to write a program without creating mathematical formulas with enough precision..

 

I solve “unsolvable problems” with ease. A common example is dividing by zero. Any division by a variable must be checked before executing the code, otherwise an exception is raised (program crash and shutdown will occur).

I actually would prefer the formula, but if a computer program would help to get that formula, that would be great. Unfortunately, I am not a programmer to do that.

[Sensei]

So basically what is the purpose of this?

51 minutes ago, tmdarkmatter said:

but if a computer program would help to get that formula, that would be great.

The computer program will not give the formula, but it will give the value of the result..

I.e., you enter the radius of the sphere, the position of P, and some data about the slope, and the program will calculate the rest by itself.

For example, you want to calculate the area under some curve.

You can draw these figures using graphics functions and then count the number of pixels. With ease.

If you have "step between pixels", treat it as 1/2.

Repeat this for the Z-axis and you get the volume.

Graphics cards now have thousands of parallel GPU cores.

Handmade brute-force integration.

The higher the resolution, the better the precision.

 

 

If it's “for fun,” why should we do it for you? You should have fun coming up with the formula yourself. If we do it for you, you won't have fun.. And us too, because we will be used.

 

Blender or another 3D application should give you an approximation - create a 3D sphere, create a cone, subtract these objects using the boolean tools, and the second time cut these objects and use the volume tool. After a few tries with a different slope of the cone, you will be well on your way to getting an approximate result.

Edited 22 hours ago by Sensei

[Sensei]

10 hours ago, tmdarkmatter said:

Another problem is that the area where the cone is touching the sphere is not even flat, so this would even include the calculation of two domes on both sides.

The volume (VOLUME! V=4/3 pi r^3) of the sphere / 2 (divided by two) must be equal to the volume of the dome on the left (from the 2D image) plus the volume of the dome on the right plus the volume of the cone with the right part cut off (the volume of the cone with height “a+b” minus the volume of the cone with height “a”).

We do not know (yet) the values of a and b, but we know that the distance from the vertex of the cone to the sphere is 4 (5-1).

 

 

[Genady]

2 hours ago, tmdarkmatter said:

The problem is that all of these volumes still have an unknown value for "b" that still has to be calculated.

This is one unknown, and you get one equation. Then solve the equation.

[tmdarkmatter]

4 hours ago, Sensei said:

So basically what is the purpose of this?

Yes, I understand what you mean. I will try to create an equation because I actually need the equation. It´s just part of a personal project. If I will not have a good equation during the following weeks, I will try to use Blender for this to get an approximation. In a worst case scenario, I might just have a chart with a list of different distances d and the corresponding values for the distance between the intersections.

 

3 hours ago, Sensei said:

We do not know (yet) the values of a and b, but we know that the distance from the vertex of the cone to the sphere is 4 (5-1).

At least we know that a is -b/d and the distance from the vertex to the intersection with the sphere is a little more than 4 and it becomes less if the point p gets closer to the sphere because the angle would increase, not only because the distance would decrease.

Edited 18 hours ago by tmdarkmatter

[studiot]

Firstly can we be assured that this is neither for a competition, homework or coursework ?

 

Given that,

A cone is surface generated by a pencil of lines passing through a fixed point and intersecting a fixed closed curve.

There is a closed form analytical formula for the cone in 3D when that curve is a conic.

Since a conci is a plane curve, you can take advantage of that by splitting the cone into two parts along the lines that genady has already mentioned.

You need to make your splitting plane parallel to the Z axis.

Then the conic is a circle in the xy plane.

The radius may then be found by using the fact that the radius equals the Y coordinate at Z = 0, in this plane.

Ref:

Longchamps

Problemes de Geometrie Analytique  vol3