[Maths] Pyramid Equotation

[Yixx]

Hello people,,

I am in need of an equotation for a pyramid. There are a few parameters that i have got, and some conditions.

I need a pyramid with 4 sides, all with the same dimensions.
I got 3 coordinates, X,Y and Z. And i got a range.

Now i need a pyramid that has 4 points that all have a distance of <RANGE> to point (X,Y,Z). That means that all 6 ribs of the pyramid have the same length.

The pyramid must look like this. It actually has alot in common with the CH₄ molecule.


The equotation i need is one to calculate the coordinates (all 3) of the 4 hook-points of the pyramid.

I hope some math genius will come by and tell me =)

Thanks

-Yixx,,-

 

[Mechanical Man]

So it's a tetraeder? But you can rotate those xyz points as you wish?

 

[Yixx]

Well,, i kinda got like a 'facing' of the pyramid. Lets keep that one at 0 for now, then i just need the equotations to calculate the other points.

the upper point is easy, just x_center,y_center,z_center+Range

But the others?

 

[Mechanical Man]

At least two point must be fixed.

 

[Yixx]

The center point and the upper point are fixed,, that actually ruins the concept of a randomly spinning (in all axes') pyramid, but i'll just figure something out,,

So: the center and the upper point are fixed,, and then?

 

[risk_fan2]

You need to fix the center and 2 vertices, or else there are an unlimited number of possible Tetrahedrons...

 

[GhostWolf]

Not entirely sure if the axes are set correctly since I am used to normal axes (where z is depth, not height), but it shouldn't be too hard to fix if it isn't correct.

Anyway, there are probably a million syntax errors since I didn't use Jass in who knows how long and I don't have any warcraft related programs, but I think this should do the job if fixed.

Basically just create a Pyramid instance with a Vector3 resembling the center, a real being the length (distance of each point from the center), and a real being the angle of the rotation from the center to the four corners.
The angle would probably be best at -45 degrees (give it as radians to the function).

library LibPyramid

struct Vector3
    public real x
    public real y
    public real z
    
    public method create takes real x,real y,real z returns nothing
        this = Vector3.allocate()
        set .x = x
        set .y = y
        set .z = z
    endmethod
    
    public method operator= takes Vector3  sV returns nothing
        set .x = sV.x
        set .y = sV.y
        set .z = sV.z
    endmethod

    private method squaredLength takes nothing returns real
        return ((.x*.x)+(.y*.y)+(.z*.z))
    endmethod
    
    private method normalize takes nothing returns nothing
        local real length = squaredLength;
        
        if length > 1 then
            set length = SquareRoot(length)
            
            set .x = .x / length
            set .y = .y / length
            set .z = .z / length
        endif
    endmethod
    
    // not suggested for use (which is why I set it to private)
    // use rotateNormalized instead (see below)
    private method rotate takes real Radians,real AxisX,real AxisY,real AxisZ returns Vector3
        
        local Vector3 ret = Vector3.create()
        
        local real fCos = Cos(Radians)
        local real fSin = Sin(Radians)
        local real fOneMinusCos = 1.0-fCos
        local real fX2 = AxisX*AxisX;
        local real fY2 = AxisY*AxisY;
        local real fZ2 = AxisZ *AxisZ ;
        local real fXYM = AxisX*AxisY*fOneMinusCos;
        local real fXZM = AxisX*AxisZ *fOneMinusCos;
        local real fYZM = AxisY*AxisZ *fOneMinusCos;
        local real fXSin = AxisX*fSin;
        local real fYSin = AxisY*fSin;
        local real fZSin = AxisZ *fSin;
    
        local real m00 = fX2*fOneMinusCos+fCos;
        local real m01 = fXYM-fZSin;
        local real m02 = fXZM+fYSin;
        local real m10 = fXYM+fZSin;
        local real m11 = fY2*fOneMinusCos+fCos;
        local real m12 = fYZM-fXSin;
        local real m20 = fXZM-fYSin;
        local real m21 = fYZM+fXSin;
        local real m22 = fZ2*fOneMinusCos+fCos;

        set ret.x = (.x*m00)+(.y*m10)+(.z*m20)
        set ret.y = (.x*m01)+(.y*m11)+(.z*m21)
        set ret.z = (.x*m02)+(.y*m12)+(.z*m22)
        
        return ret
    endmethod
    
    public method rotateNormalized takes real Radians,real AxisX,real AxisY,real AxisZ returns Vector3
        Axis.normalize()
        .rotate(Radians,Axis)
    endmethod
endstruct

struct Pyramid
    public Vector3 Center
    public Vector3 Top
    public Vector3 A
    public Vector3 B
    public Vector3 C
    public real Length
    public real Angle
    
    public method create takes Vector3 Center,real Length,real Angle returns nothing
        local Vector3 Unit = Vector3.create(1,0,0)
        
        set .Center = Vector3.create(Center.x,Center.y,Center.z)
        set .Top = Vector3.create(Center.x,Center.y,Center.z + Length)
        
        set Unit = Unit.rotateNormalized(Angle,1,0,0)
        
        set Unit = Unit.rotateNormalized(120*bj_DEGTORAD,0,0,1)
        set .A = Vector3.create(Unit .x * Length,Unit .y * Length,Unit .z * Length)
        
        set Unit =  Unit.rotateNormalized(120*bj_DEGTORAD,0,0,1)
        set .B = Vector3.create(Unit .x * Length,Unit .y * Length,Unit .z * Length)
        
        set Unit =  Unit.rotateNormalized(120*bj_DEGTORAD,0,0,1)
        set .C = Vector3.create(Unit .x * Length,Unit .y * Length,Unit .z * Length)
        
        set .Length = Length
        set .Angle = Angle
    endmethod
    
endstruct

endlibrary

 

[Deaod]

Alright, lets assume you the point M given. Point A is straight above point M, so A(x|y|z+r).

Because were talking about a regular tetraeder, the length of each edge of the tetraeder (a) is sqrt(2/3)*2*range.
Edit: wtf am i talking. Its the length of the edges of a cube surrounding the tetraeder so that each edge of the tetraeder is a planar diagonal.

The second Point B is at 0 degrees, so B( x+((a/2)*Cos(0) | y+((a/2)*Sin(0)) | z-(r/2) ).

The third and fourth point are at 2*Pi/3 and 4*Pi/3.

 

[GhostWolf]

Right, three points. Lol.

 

[Yixx]

Ok im back.
So lets say the 'facing' of the pyramid is 0. And one point is always on the pyramids facing.
then one point would be P(x+Range*Cos(0) | y + Range*Sin(0) | z - Range*Sin(109.5-90) )

Why the 109.5? That is the exact angle in a CH₄ from each H atom to an other H atom. Now the the -90 is for a reason i cannot explain very well in english but i hope you understand.

Now every of the next points will be:
P2(x+Range*Cos(120) | x + Range*Sin(120) | z+Range*Sin(109.5-90) )
P3(x+Range*Cos(240) | x + Range*Sin(240) | z+Range*Sin(109.5-90) )
120 + facing, but facing = 0
240 + facing, but facing = 0

That is when the upper point is fixed, now how do i do those point when the upper is non-fixed?

BTW: Those vectors might be usefull!

 

[GhostWolf]

I wonder if I should post vector, matrix and quaternion libraries just for the lulz, since I am making them in C++ anyway.
Not that they are really useful for anything in warcraft xD

 

[Yixx]

Yes post them!,, But tell me, am i right above about the points?

 

[GhostWolf]

I have no idea, but do notice that you need to convert everything there to radians.

I think this really is easy to make with vectors, since the same code works for every rotations without any edits.
Something like this (now it probably looks like I am bragging).

 

[Yixx]

Ok i got a few new things:

struct DATA
    real x
    real y
    real z
    real Range
    real zRoll
    real xyRoll // = facing
endstruct

Now it seems to me that this will be the 5 points:

Center ( x | y | z )
Upper ( x + (Range*Sin(zRoll)) * Cos(xyRoll) | y + (Range*Sin(zRoll)) * Sin(xyRoll) | z + Range*Cos(zRoll) )
Lower₁ ( x + (Range * Cos(19.5))*Cos(000+xyRoll) | x + (Range * Cos(19.5))*Sin(000+xyRoll) | z - Range* Sin(19.5) )
Lower₂ ( x + (Range * Cos(19.5))*Cos(120+xyRoll) | x + (Range * Cos(19.5))*Sin(120+xyRoll) | z - Range* Sin(19.5) )
Lower₃ ( x + (Range * Cos(19.5))*Cos(240+xyRoll) | x + (Range * Cos(19.5))*Sin(240+xyRoll) | z - Range* Sin(19.5) )


Please tell me if i am wrong or not =\

EDIT:
Current Code:

It works, forgot to convert to radians ^^,,

Code

library Pyramid initializer Init 
    globals
        constant integer C =0
        constant integer U =1
        constant integer L1=2
        constant integer L2=3
        constant integer L3=4
        constant real Cos195 = Cos(19.5*bj_DEGTORAD)
        constant real Sin195 = Sin(19.5*bj_DEGTORAD)
    endglobals
 
 
    struct DATA
        real array x [5]
        real array y [5]
        real array z [5]
        lightning array L [5]
        real xyRoll
        real zRoll
        real Range
 
        static method create takes real x, real y, real z returns DATA
            local DATA dat = DATA.allocate()
            local real L
            local real S
            local integer i = 0
            set dat.xyRoll = 0
            set dat.zRoll = 0
            set dat.Range = 500
            set dat.x[C] = x
            set dat.y[C] = y
            set dat.z[C] = z
            set L = dat.Range*Sin(dat.zRoll)
            set dat.x[U] = x + L * Cos(dat.xyRoll)
            set dat.y[U] = y + L * Sin(dat.xyRoll)
            set dat.z[U] = z + dat.Range * Cos(dat.zRoll)
            set L = dat.Range * Cos195
            loop
                exitwhen i > 2
                set dat.x[i+2] = x + L * Cos(120*i + dat.xyRoll)
                set dat.y[i+2] = y + L * Sin(120*i + dat.xyRoll)
                set dat.z[i+2] = z - dat.Range * Sin195
                set i = i + 1
            endloop
            set dat.L[0] = AddLightningEx("CLPB",false,dat.x[U],dat.y[U],dat.z[U],dat.x[L1],dat.y[L1],dat.z[L1])
            set dat.L[1] = AddLightningEx("CLPB",false,dat.x[U],dat.y[U],dat.z[U],dat.x[L2],dat.y[L2],dat.z[L2])
            set dat.L[2] = AddLightningEx("CLPB",false,dat.x[U],dat.y[U],dat.z[U],dat.x[L3],dat.y[L3],dat.z[L3])
            set dat.L[0] = AddLightningEx("CLPB",false,dat.x[L1],dat.y[L1],dat.z[L1],dat.x[L2],dat.y[L2],dat.z[L2])
            set dat.L[0] = AddLightningEx("CLPB",false,dat.x[L1],dat.y[L1],dat.z[L1],dat.x[L3],dat.y[L3],dat.z[L3])
            set dat.L[0] = AddLightningEx("CLPB",false,dat.x[L2],dat.y[L2],dat.z[L2],dat.x[L3],dat.y[L3],dat.z[L3])
            return dat
        endmethod
    endstruct
    private function Start takes nothing returns nothing
        call DATA.create(0,0,300)
    endfunction
    private function Init takes nothing returns nothing
        local trigger t = CreateTrigger()
        call TriggerRegisterTimerEvent(t,1,false)
        call TriggerAddAction(t,function Start)
    endfunction
endlibrary

But it is not working as it should