Math problem revisited.

[NOVARSEG]

https://www.qb64.org/forum/index.php?topic=2714.30

A  triangle (call it the base triangle) is formed by 3 points x, y, z   Find if a 4th  point x, y ,z rests on the triangle or not.

I find  @Ashish solution interesting.  However the whole thread (I think) is dealing with 2D and I thought Ashish's solution would work for 3D as well.

Consider a 4th point not on the triangle.  There are 4 points total x, y, z which form a tetrahedron.    4  triangles form the tetrahedron.  If the 4th point is not on the base triangle then the total area of the 3 remaining triangles will be greater than the area of the base triangle.

Need some math to prove that.   It looks like the 2D case was proven already so it should work for 3D .

[NOVARSEG]

Some code

In the code example, the 4th point is not on the triangle.?

Code: QB64: [Select]

1. TYPE V3
2.     X AS SINGLE
3.     Y AS SINGLE
4.     Z AS SINGLE
5. END TYPE
6.  
7. DIM P1 AS V3
8. DIM P2 AS V3
9. DIM P3 AS V3
10. DIM P4 AS V3
11.  
12. P1.X = 10: P1.Y = 2: P1.Z = 1 'base triangle
13. P2.X = 9: P2.Y = 6: P2.Z = 7 'base triangle
14. P3.X = 3: P3.Y = 4: P3.Z = 3 'base triangle
15. P4.X = 1: P4.Y = 3: P4.Z = 2 'point to test
16.  
17. DISTANCE P1, P2, d1 'side lengths of base triangle
18. DISTANCE P1, P3, d2 'side lengths of base triangle
19. DISTANCE P2, P3, d3 'side lengths of base triangle
20.  
21. DISTANCE P1, P4, d4 'other side lengths of tetrahedron
22. DISTANCE P2, P4, d5 'other side lengths of tetrahedron
23. DISTANCE P3, P4, d6 'other side lengths of tetrahedron
24.  
25.  
26. AREA d1, d2, d3, a1 'base triangle
27. AREA d1, d4, d5, a2 'other
28. AREA d2, d6, d4, a3 'other
29. AREA d3, d5, d6, a4 'other
30.  
31.  
32. PRINT a1
33. PRINT a2 + a3 + a4
34.  
35.  
36.  
37. 'Heron's formula for area of a triangle
38. SUB AREA (var1, var2, var3, var4 AS SINGLE)
39.     ta = var1 + var2 + var3
40.     var4 = (ta + (ta - var1) + (ta - var2) + (ta - var3)) ^ .5
41. END SUB
42.  
43.  
44. SUB DISTANCE (Var1 AS V3, var2 AS V3, var3 AS SINGLE)
45.     var3 = ((Var1.X - var2.X) ^ 2 + (Var1.Y - var2.Y) ^ 2 + (Var1.Z - var2.Z) ^ 2) ^ .5
46. END SUB

the dotty stuff is OK

I can't get the 4th point on the triangle.  Testing with a 3,4,5 right triangle to see where the math went wrong.

I'm getting the correct area for a 3,4,5 triangle but the other areas are not correct.

[MasterGy]

Hello ! i just looked at the area calculation and there may be 2 things wrong. I’m not sure, but where I found the heron formula, I rewrote it that way. See if that works.

Code: QB64: [Select]

1. SUB AREA (var1, var2, var3, var4 AS SINGLE)
2.     ta = (var1 + var2 + var3) / 2
3.     var4 = (ta * ((ta - var1) + (ta - var2) + (ta - var3))) ^ .5
4. END SUB
5.  

[NOVARSEG]

@MasterGy

 Yes that was where the error was. Good catch.

 I watched a video on Heron's formula and got it totally wrong.  Here is updated code it seems to work now.

It shows a 3 ,4, 5 triangle with z = 0  and the print out shows equal areas, so point  4 in this case is on the base triangle. 

I'm going to put some iterations in the next program so if, say, x and y are known for point 4 then it will calculate z etc.

Code: QB64: [Select]

1. TYPE V3
2.     X AS SINGLE
3.     Y AS SINGLE
4.     Z AS SINGLE
5. END TYPE
6.  
7. DIM P1 AS V3
8. DIM P2 AS V3
9. DIM P3 AS V3
10. DIM P4 AS V3
11.  
12. P1.X = 0: P1.Y = 0: P1.Z = 0 'base triangle
13. P2.X = 0: P2.Y = 4: P2.Z = 0 'base triangle
14. P3.X = 3: P3.Y = 0: P3.Z = 0 'base triangle
15. P4.X = 1: P4.Y = 1: P4.Z = 0 'point to test
16.  
17. DISTANCE P1, P2, d1 'side lengths of base triangle
18. DISTANCE P1, P3, d2 'side lengths of base triangle
19. DISTANCE P2, P3, d3 'side lengths of base triangle
20.  
21. DISTANCE P1, P4, d4 'other side lengths of tetrahedron
22. DISTANCE P2, P4, d5 'other side lengths of tetrahedron
23. DISTANCE P3, P4, d6 'other side lengths of tetrahedron
24.  
25. AREA d1, d2, d3, a1 'base triangle
26. AREA d1, d4, d5, a2 'other
27. AREA d2, d4, d6, a3 'other
28. AREA d3, d5, d6, a4 'other
29.  
30. PRINT a1
31. PRINT a2 + a3 + a4
32.  
33. 'Heron's formula for area of a triangle
34. SUB AREA (var1, var2, var3, var4 AS SINGLE)
35.     ta = (var1 + var2 + var3) / 2
36.     var4 = (ta * (ta - var1) * (ta - var2) * (ta - var3)) ^ .5
37. END SUB
38.  
39. SUB DISTANCE (Var1 AS V3, var2 AS V3, var3 AS SINGLE)
40.     var3 = ((Var1.X - var2.X) ^ 2 + (Var1.Y - var2.Y) ^ 2 + (Var1.Z - var2.Z) ^ 2) ^ .5
41. END SUB
42.  

[Ashish]

Hmmmmm...... I will post something here in few days.