# 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
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:
¯¯¯¯¯¯¯
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 ⎭ ⎭

¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
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 · Î± [rad] = 6385,112```