Constructing Objects In Point World
Animation1 shows how a Sphere Object might look in Point World. It was constructed
using 126 individual Points which are each drawn as a small Ball. In Point
World the small Balls represent actual Points. Each Ball is constructed
using 72 flat polygons. The shading effect works on the individual polygons to create
the overall shading of the Ball. Balls are just markers for where
a Point is located. Balls can be drawn packed in more tightly or more loosely.
Balls can be packed so tight that they overlap but they still represent
Points. The actual diameter of a Ball is not important just the central
Point location that it represents is important. The fundamental Objects in Point World are the Line, the Circle, and the Sphere. Each of these Objects can be Empty or Full.
- Empty Line: Points located at the two ends of the Line.
- Empty Circle: Points located at discrete locations on the circumference of the Circle.
- Empty Sphere: Points located at discrete locations on the surface of the Sphere.
- Full Line: Points located at discrete locations over whole length of the Line.
- Full Circle: Points located at discrete locations over whole area of the Circle.
- Full Sphere: Points located at discrete locations over whole volume of the Sphere.
Note that the equations in this section are for people who need to see the details. It is not necessary to understand the equations to understand the different configurations that the Animations portray. The important thing to take away from this section is that Spheres and Circles are sometimes constructed in an odd looking way that makes the Animations for rotations easier to understand.
The size of the Balls that represent Points is variable, and the size of the Balls determine the number of Points that are needed to represent an Object. For example the following equations can be used to construct a Full Line:
NumberOfBalls = (LineLength + BallSize) / BallSize
if(NumberOfBalls is Even) NumberOfBalls = NumberOfBalls - 1
BallOffset = LineLength / (NumberOfBalls - 1)
BallLocation = n * BallOffset
0 <= n <= NumberOfBalls
So if we are constructing a 4 Unit long Full Line and the Ball size is 0.5 Unit then there will be a Ball located at each endpoint and 7 Balls offset from each other by 0.5 Units for a total of 9 Balls ( 9 Points) that represent the Full Line. The number of MPs will be decremented to an odd number if the Ball Size leads to an even number of Points. In this way there is always a Point at the center of a Full Line. The location of this central Point is considered to be the location of a Full Line or an Empty Line even though an Empty Line does not have an actual Point at the center. In Point World a Line can have a Radius that is defined to be the distance from the center to one of the endpoints, which is LineRadius = LineLength / 2.
In Point World Empty Circles can be constructed from Empty Lines, and Full Circles can be constructed from Full Lines. Lines can be used to construct a Circle with Equal Offsets or with Equal Angles as shown in the following Animations for Empty Circle construction and for Full Circle Construction:
(Animation2)
Constructing Empty Circles From Empty Lines
(Animation3)
Constructing Full Circles From Full Lines
Note that these construction techniques produce Circles with a rotational symetry around the z-Axis. The construction can also be implemented for x-Axis or y-Axis symetry. This orientation characteristic will prove to be helpful when viewing the Circle rotation Animations later.
Equations For Equal Offset (Full or Empty) Circle Construction:
NumberOfLines = (2 * CircleRadius + BallSize) / BallSize
if(NumberOfLines is Even) NumberOfLines = NumberOfLines - 1
Span = NumberOfLines / 2
DeltaOffset = CircleRadius / (NumberOfLines - 1)
LineOffset = n * DeltaOffset
Angle = asin(LineOffset / CircleRadius)
LineRadius = CircleRadius * cos(Angle)
-Span <= n <= Span
Equations For Equal Angle (Full or Empty) Circle Construction:
BallAngleSize = 2.0 * asin((BallSize / 2.0) / CircleRadius)
NumberOfLines = (Pi + BallAngleSize) / BallAngleSize
if(NumberOfLines is Even) NumberOfLines = NumberOfLines - 1
Span = NumberOfLines / 2
DeltaAngle = Pi / (NumberOfLines - 1)
Angle = n * DeltaAngle
LineOffset = CircleRadius * sin(Angle)
LineRadius = CircleRadius * cos(Angle)
-Span <= n <= Span
These equations will produce Points that have CircleRadius as follows:
CircleRadius = sqrt(LineOffset^2 + LineRadius^2)
Full
Circles can also be constructed from Concentric Empty Circles with Equal Offsets or with Equal Angles as shown in the following Animation:
(Animation4)
Constructing Full Circles From Concentric Empty Circles
Equations For Concentric Full Circle Construction:
NumberOfConcentricCircles = (CircleRadius + BallSize) / BallSize;
if ((NumberOfConcentricCircles % 2) == 0) NumberOfConcentricCircles = NumberOfConcentricCircles - 1
DeltaRadius = CircleRadius / (NumberOfConcentricCircles - 1)
ConcentricCircleRadius = n * DeltaRadius
0 <= n <= NumberOfConcentricCircles
In Point World Empty Spheres can be constructed from Empty Circles, and Full Spheres can be constructed from Full Circles. Full Spheres are constructed from Full Circles that are constructed from Concentric Circles. Circles can be used to construct a Sphere with Equal Offsets or with Equal Angles as shown in the following Animations for Empty Sphere construction and for Full Sphere construction:
(Animation5)
Constructing Empty Spheres From Empty Circles
(Animation6)
Constructing Full Spheres From Full Circles
Note that these construction techniques produce Circles with a rotational symetry around the z-Axis. The construction can also be implemented for x-Axis or y-Axis symetry. This orientation characteristic will prove to be helpful when viewing the Circle rotation Animations later.
Note that these construction techniques produce Spheres that have a rotational symetry around the y-Axis. The construction can also be implemented for x-Axis or y-Axis symetry. This rotational symetry characteristic will prove to be helpful when viewing the Sphere rotation Animations later.
Equations For Equal Offset (Full or Empty) Sphere Construction:
NumberOfCircles = (2 * SphereRadius + BallSize) / BallSize
if(NumberOfCircles is Even) NumberOfCircles = NumberOfCircles - 1
Span = NumberOfCircles / 2
DeltaOffset = SphereRadius / (NumberOfCircles - 1)
CircleOffset = n * DeltaOffset
Angle = asin(CircleOffset / SphereRadius)
CircleRadius = SphereRadius * cos(Angle)
-Span <= n <= Span
Equations For Equal Angle (Full or Empty) Sphere Construction:
BallAngleSize = 2.0 * asin((BallSize / 2.0) / SphereRadius)
NumberOfCircles = (Pi + BallAngleSize) / BallAngleSize
if(NumberOfCircles is Even) NumberOfCircles = NumberOfCircles - 1
Span = NumberOfCircles / 2
DeltaAngle = Pi / (NumberOfCircles - 1)
Angle = n * DeltaAngle
CircleOffset = SphereRadius * sin(Angle)
CircleRadius = SphereRadius * cos(Angle)
-Span <= n <= Span
These equations will produce Points that have SphereRadius as follows:
SphereRadius = sqrt(CircleOffset^2 + CircleRadius^2)
In Point World Empty Hyper Spheres can be constructed from Empty Spheres, and Full Hyper Spheres can be constructed from Full Spheres. Spheres can be used to construct a Hyper Sphere with Equal Offsets or with Equal Angles. A Hyper Sphere cannot be depicted directly. Hyper Spheres can only be indirectly depicted using Axis Sharing which will be explained later. The following Animation shows how Hyper Spheres might look using Axis Sharing:
(Animation7)
Constructing Hyper Spheres From Spheres
Equations For Equal Offset (Full or Empty) Hyper Sphere Construction:
NumberOfSpheres = (2 * HyperSphereRadius + BallSize) / BallSize
if(NumberOfSpheres is Even) NumberOfSpheres = NumberOfSpheres - 1
Span = NumberOfSpheres / 2
DeltaOffset = HyperSphereRadius / (NumberOfSpheres - 1)
SphereOffset = n * DeltaOffset
Angle = asin(SphereOffset / HyperSphereRadius)
SphereRadius = HyperSphereRadius * cos(Angle)
-Span <= n <= Span
Equations For Equal Angle (Full or Empty) HyperSphere Construction:
BallAngleSize = 2.0 * asin((BallSize / 2.0) / HyperSphereRadius)
NumberOfSpheres = (Pi + BallAngleSize) / BallAngleSize
if(NumberOfSpheres is Even) NumberOfSpheres = NumberOfSpheres - 1
Span = NumberOfSpheres / 2
DeltaAngle = Pi / (NumberOfSpheres - 1)
Angle = n * DeltaAngle
SphereOffset = HyperSphereRadius * sin(Angle)
SphereRadius = HyperSphereRadius * cos(Angle)
-Span <= n <= Span
These equations will produce Points that have HyperSphereRadius as follows:
HyperSphereRadius = sqrt(SphereOffset^2 + SphereRadius^2)
Note that these equations will produce Points that have ObjRadius. This can be seen by recognizing that the Offset is on the Hyper Axis (y or w) and the Radius is perpendicular to the Hyper Axis regardless of the particular Point. It is easy in 3D space to see how all the Points of a component 2D Circle are at a Radius that is perpendicular to the y-axis. In 4D space it is more difficult to see how all the Points on a component 3D Sphere can be perpendicular to the w-Axis. It must be realized that relative to 4D space our 3D space is flat and the 3D Sphere is also flat. The whole 3D space is perpendicular to the 4D w-Axis so the Points of the 3D Sphere in 4D are all on the same 3D plane and therefore the Radius lines are all perpendicular to the w-Axis. So the Radius line and the Offset line define a right triangle in the n-dimensional space and it is easy to see that the line from the Point to the origin completes the triangle and is the distance of the Point from the origin which is ObjRadius. So:
ObjRadius = sqrt(Offset^2 + Radius^2)
It is easy to see how 2D Circles are made out of Point Pairs and how 3D Spheres are made out of 2D Circles but it is more difficult to see how Hyper Spheres are made out of 3D Spheres because we can not ever really see a Hyper Sphere. We can only theorize about it and see it using Axis Sharing or some other method. We actually need a different kind of Brain. We need a 4D Brain and Visual System. Similarly a 2D Point Man can not ever really see a 3D Sphere. He needs a 3D Brain and Visual System. Even as 3D beings we can not show a 2D Point Man how to see a 3D Sphere. A higher dimensional space is not just more of the lower dimensional space but rather it is a completely different thing. Still, more can be learned about 4D space especially when it comes to rotations as will be shown later.