MatheAss 10.0 − 3‑dim. Geometry
Coordinate systems
With this program, three‑dimensional Cartesian coordinates can be converted into spherical or cylindrical 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 solids
The program calculates the five Platonic solids — tetrahedron, hexahedron, octahedron, dodecahedron and icosahedron — when edge length, face height, space height, inradius, circumradius, volume or surface area are given.
Tetrahedron
Given:
¯¯¯¯¯
Circumradius rc = 1
Results:
¯¯¯¯¯¯
Edge a = 1,6329932
Apothem h1 = 1,4142136
Altitude h2 = 1,3333333
Inradius ri = 0,33333333
Volume V = 0,51320024
Surface S = 4,6188022
Hexahedron
Given:
¯¯¯¯¯
Inradius ri = 1
Results:
¯¯¯¯¯¯
Edge a = 2
Face Diagonal d1 = 2,8284271
Body Diagonal d2 = 3,4641016
Circumradius rc = 1,7320508
Volume V = 8
Surface S = 24
Octahedron
Given:
¯¯¯¯¯
Inradius ri = 1/sqrt(3)
Results:
¯¯¯¯¯¯
Edge a = 1,4142136
Apothem h1 = 1,2247449
Altitude h2 = 2
Circumradius rc = 1
Volume V = 1,3333333
Surface S = 6,9282032
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
Icosahedron
Given:
¯¯¯¯¯
Altitude h2 = 2
Results:
¯¯¯¯¯¯
Edge a = 1,0514622
Apothem h1 = 0,910593
Circumradius rc = 1
Inradius ri = 0,79465447
Volume V = 2,5361507
Surface S = 9,5745414
Other solids
The program calculates all quantities of a regular prism, a right circular cylinder, a square pyramid, a right circular cone or a sphere, when two of them are given.
The prism
Given:
¯¯¯¯¯
Vertices n = 6
Circumradius rc = 1
Volume V = 1
Results:
¯¯¯¯¯¯
Edge a = 1
Altitude h = 0,38490018
Inradius ri = 0,8660254
Base B = 2,5980762
Surface S = 7,5055535
The circular cylinder
Given:
¯¯¯¯¯
Radius r = 1
Volume V = 1
Results:
¯¯¯¯¯¯
Altitude h = 0,31830989
Perimeter p = 6,2831853
Base B = 3,1415927
Lateral Surface L = 2
Surface S = 8,2831853
The square pyramid
Given:
¯¯¯¯¯
Edge a = 1
Volume V = 1
Results:
¯¯¯¯¯¯
Lateral edge s = 3,082207
Altitude h1 = 3
Apothem h2 = 3,0413813
Surface S = 7,0827625
Face A = 1,5206906
The circular cone
Given:
¯¯¯¯¯
Altitude h = 1
Surface S = 6
Results:
¯¯¯¯¯¯
Radius r = 0,86994247
Apothem s = 1,3254433
Volume V = 0,79251901
Lateral Surface L = 3,622443
Base B = 2,377557
The sphere
Given:
¯¯¯¯¯
Surface S = 1
Results:
¯¯¯¯¯¯
Radius r = 0,28209479
Diameter d = 0,56418958
Perimeter p = 1,7724539
Volume V = 0,094031597
Parallel circle α = 48°
Radius r' = 0,18875826
Perimeter u' = 1,1860031
Line through 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
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Orthogonal projection: 4·x - y = 3
Point of intersection: S1(0,8|0,2|0)
Angle of intersection: 50,490288°
Position to the yz plane
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Orthogonal projection: 5·y – 4·z = 1
Point of intersection: S2(0|-3|-4)
Angle of intersection: 8,8763951°
Position to the xz plane
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Orthogonal projection: 5·x - z = 4
Point of intersection: S3(0,75|0|-0,25)
Angle of intersection: 38,112927°
Plane through 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 through 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 space
The program calculates intersections of lines, planes and spheres.
two 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)
Angle of intersection: 60°
Distances from the origin:
d(O,g)=5 d(O,h)=5
plane and line
-> ⎧ 5 ⎫ ⎧ 0 ⎫
g : x = ⎪ 0 ⎪ + r·⎪ 1 ⎪
⎩ 0 ⎭ ⎩ 1 ⎭
E : x + y + z = 5
Intersection point :
S(5|0|0)
Angle of intersection:
α = 54.73561°
sphere and line
-> ⎧ 1 ⎫ ⎧ 1 ⎫
g : x = ⎪ 0 ⎪ + r·⎪ 1 ⎪
⎩ 0 ⎭ ⎩ 1 ⎭
K : M(5|5|5) , r = 5
Intersection points :
S1(2.8187|1.8187|1.8187)
S2(8.5147|7.5147|7.5147)
Chord length :
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:
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
α = 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(-0,1363636|2,909091|1,863636)
r = 3,5483671
Distances on the sphere
(New in version 9.0 from December 2021)
The program calculates the distance between two points on a sphere. Several functions of MatheAss are combined for this purpose.
Radius of the sphere ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ r = 6371 GPS decimal ¯¯¯¯¯¯¯¯¯¯¯ Berlin :52.523403, 13.4114 New York :40.714268, -74.005974 GPS dms ¯¯¯¯¯¯¯ Berlin :52°31'24.2508"N, 13°24'41.04"E New York :40°42'51.3648"N, 74°0'21.5064"W ... Distance ¯¯¯¯¯¯¯¯ d = r · α [rad] = 6385,112
