MatheAss 9.0 − 3-dim. Geometry

Coordinate systems

With this program, three-dimensional Cartesian coordinates can be converted into spherical coordinates or cylinder coordinates and vice versa.


cartesian            polar                           cylindrical
   x  =  1              r  =  1.7320508           ρ  =  1.4142136
   y  =  1             φ  =  45°                      φ  =  45°  
   z  =  1             Θ =  35,26439°            z  =  1      

Platonic bodies

The program calculates the five Platonic bodies tetrahedron, hexahedron, octahedron, dodecahedron and icosahedron if edge length, surface height, room height, insphere radius, umbelly radius, volume or surface are given.

Example: Dodecahedron

Given:
¯¯¯¯¯¯
   Face Diagonal d = 2

Results:
¯¯¯¯¯¯¯
                  Edge a = 1,236068
      Face Altitude h = 1,902113
    Circumradius rc = 1,7320508
             Inradius ri = 1,3763819
              Volume V = 14,472136
             Surface S = 31,543867

Other bodies

The program calculates all the sizes of a regular prism, a vertical circular cylinder, a regular pyramid, a vertical circular cone or a sphere if two of them are given.

Example: Circular cone

Given:
¯¯¯¯¯¯
            Volume V = 1
                Base B = 1

Results:
¯¯¯¯¯¯¯
              Radius r = 0,56418958
             Altitude h = 3
           Apothem s = 3,0525907
 Lateral Surface L = 5,4105761
             Surface S = 6,4105761

Straight Line by 2 points

Line through  A(1|1|1), B(2|5|6)

Parametric representation
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
  ->  ⎧ 1 ⎫     ⎧ 1 ⎫
  x = ⎪ 1 ⎪ + t·⎪ 4 ⎪
      ⎩ 1 ⎭     ⎩ 5 ⎭

Distance from origin
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
  d = 0,78679579

Position to the xy plane
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
 Projection  : 4·x - y = 3
 Inters.Point: S1(0,8|0,2|0)
 Inters.Angel: 50,490288°

Position to the yz plane
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
 Projection  : 5·y - 4·z = 1
 Inters.Point: S2(0|-3|-4)
 Inters.Angel: 8,8763951°

Position to the xz plane
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
 Projection  : 5·x - z = 4
 Inters.Point: S3(0,75|0|-0,25)
 Inters.Angel: 38,112927°

Plane by 3 points

Plane through the points:
A(1|2|3), B(2|3|3), C(1|0|1)

Point-slope-form:
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
->  ⎧ 1 ⎫   ⎧ 1 ⎫   ⎧ 0 ⎫
x = ⎪ 2 ⎪+r·⎪ 1 ⎪+s·⎪ 1 ⎪
    ⎩ 3 ⎭   ⎩ 0 ⎭   ⎩ 1 ⎭

Equation in coordinates:
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
x - y + z = 2

Distance from origin:
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
d = 1,1547005

Trace points:
¯¯¯¯¯¯¯¯¯¯¯¯¯
Sx(2|0|0) 
Sy(0|-2|0) 
Sz(0|0|2)







Sphere by 4 points

Sphere through the points: 
A(1|0|0), B(0|2|0), 
C(0|0|3), D(1|0|1)

Normal form:
¯¯¯¯¯¯¯¯¯¯¯¯
     ⎧ ->  ⎧-2,5 ⎫ ⎫2 
 K : ⎪ x - ⎪-0,5 ⎪ ⎪ = 12,75
     ⎩     ⎩ 0,5 ⎭ ⎭

Center and radius:
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
 M(-2,5|-0,5|0,5)
 r = 3,5707142
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Intersections in the space

The program calculates the sections of straight lines, planes and spheres.

Two Straight Lines


    ->  ⎧ 5 ⎫     ⎧ 0 ⎫
g : x = ⎪ 0 ⎪ + r·⎪ 1 ⎪
        ⎩ 0 ⎭     ⎩ 1 ⎭

    ->  ⎧ 0 ⎫     ⎧ 1 ⎫
h : x = ⎪ 5 ⎪ + s·⎪ 0 ⎪
        ⎩ 0 ⎭     ⎩ 1 ⎭

Intersection point: S(5|5|5)

Intersection angle: 60°

Distances from origin :
  d(O,g)=5  d(O,h)=5
  
  
  

Plane and Straight Line

Plane E :
¯¯¯¯¯¯¯¯¯
  E : x + y + z = 5

Line g :
¯¯¯¯¯¯¯¯
      ->  ⎧ 5 ⎫     ⎧ 0 ⎫
  g : x = ⎪ 0 ⎪ + r·⎪ 1 ⎪
          ⎩ 0 ⎭     ⎩ 1 ⎭

Intersection point :
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
  S(5|0|0)

Intersection angle :
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
  alpha = 54,73561°
  

Sphere and Straight Line

Sphere :
¯¯¯¯¯¯¯¯
  K : M(5|5|5) ,  r = 5

Line :
¯¯¯¯¯¯
     ->  ⎧ 1 ⎫     ⎧ 1 ⎫
 g : x = ⎪ 0 ⎪ + r·⎪ 1 ⎪
         ⎩ 0 ⎭     ⎩ 1 ⎭

Intersection points :
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
S1(2,81867|1,81867|1,81867)
S2(8,51467|7,51467|7,51467)

Length of the chord :
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
  s = 9,8657657

Two Planes

Given the two planes:
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
  E1 : 5·x - 2·y = 5
  E2 : 2·x - y + 5·z = 8

Intersection line:
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
      ->  ⎧-11 ⎫     ⎧ 10 ⎫
  g : x = ⎪-30 ⎪ + r·⎪ 25 ⎪
          ⎩  0 ⎭     ⎩  1 ⎭

Distance from origin:
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
  d = 1,5057283

Intersection angle:
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
  alpha = 65,993637°

Two Spheres

Given the two spheres:
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
K1 : M1(3|3|3) ,  r1 = 3
K2 : M2(1|1|1) ,  r2 = 3

Intersection circle:
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
  M(2|2|2), r = 2,4494897 

Intersection plane :
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
  E : x + y + z = 6
  
  
  
  
  
  

Sphere and Plane

Plane :
¯¯¯¯¯¯¯
 E : 5·x - 4·y + 5·z = -3

Sphere :
¯¯¯¯¯¯¯¯
     ⎧ ->   ⎧ 1 ⎫⎫2
 K : ⎪ x  - ⎪ 2 ⎪⎪ = 16
     ⎩      ⎩ 3 ⎭⎭

Intersection circle :
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
  M(1|1|1),  r = 2
  
  
  
  
  


Distances on the sphere (New in version 9.0 from December 2021)

The distance between two points on a sphere is calculated. The program is also suitable for converting decimal degrees into degrees, minutes and seconds (dms) and vice versa.

GPS decimal
¯¯¯¯¯¯¯¯¯¯¯
  Berlin : 52.523403, 13.4114
New York : 40.714268, -74.005974

GPS dms
¯¯¯¯¯¯¯
  Berlin : 52° 31' 24.2508" N, 13° 24' 41.0400" E
New York : 40° 42' 51.3648" N, 74°  0' 21.5064" W
  .
  .
  .
  
Distance
¯¯¯¯¯¯¯¯
   d = r · α [rad] = 6385,112