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Ghost Conical Space
Michael Rivera, Florian LeGouriellec, and Greg Popelka
Math3345.01 Dr. Carol Gee
December 11, 2012
Abstract
This article develops geometry over the Ghost Conical Space denoted G2 .
Figure 1: The Ghost Cone
Conical space is an area of geometry that has intrigued members of our group from
the day it was introduced this semester. The lack of exploration in class of the conical
space left us disappointed but in hindsight, having the ability to explore conical space
and develop a geometry from scratch is the best thing that could have happened to us.
This exquisite surface presented us with many trials and due to the complexity, lead
us to deceive ourselves multiple times developing flawed geometries. Through many
hectic series of brainstorming, and countless scrapped ideas, we feel as a group that
we have finally come to an understanding of some of the many idiosyncrasies of the
conical space. We hope to articulate our understanding in a way that creates intrigue,
curiosity and confidence in the minds of those who will take the time to read this paper.
The basis for ghost conical space is the intrinsic notion of the ghost cone.
Definition 1: Let G2 represent the space created by the reflection about the vertex
of the original cone. Each end of the conical space extends to infinity as seen in Figure
1.
Note that in G2 every cone, no matter angle of the cone, has a twin cone that extends from the cone point in the opposite direction of the original cone. The addition
of the ghost cone will help visualize lines, circles, and triangles on the space as well as
give the space some interesting properties.
Let us start by becoming an ant. If the ant is walking vertically up the original
cone he will eventually hit the cone point. The cone point is troublesome because it
sometimes can be seen as a boundary, a halting point if you will. We see this point as
the beginning. If we are the bug and we hit this point we can carry on walking continuously using this notion of the ghost cone. Looking at the figure below we can see
that the ant’s path does not halt or disappear but continues along the opposite side of
the ghost cone on an infinite path. If you were to remove the cone from the image you
would observe a truly straight line that extends infinitely diagonally in each direction.
This leads us to our definition of geodesics.
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Figure 2: The Ant Example
Definition 2: A line in G2 is a geodesic if and only if the line is a geodesic on a
360◦ cone.
If viewed intrinsically we would not experience any change in direction. The world
around us would disappear beneath our feet as we pass throught he vertex and reappear
behind us. The ghost cone is a tool for visualization and althought it does exist at the
same time it does not. The ghost cone is a reflection of the original cone about the
vertex. Anything on the ghost cone is actually on the original cone itself. In effect, the
ghost cone gives us an infinite space but instead of considering a path as bouncing back
onto itself, we preserve continuity by defining the ghost cone in this way.
Aside from the geodesics that hit the cone point there are other types of geodesics
as well. A good place to start is a cone that has a generating angle of less than 180
degrees. If we start with our flattened model and use a straight edge to draw a line we
can observe what happens to the line once we form the cone.
Definition 3: A geodesic on a N◦ cone where 0 < N < 180 will intersect itself with a
resulting angle of N◦ .
Let us start with the flattened model, draw a line that does not intersect the vertex,
and see what happens once we create our cone. A generating angle of 180◦ results in
geodesics which should intersect itself at an angle of 180◦. This is impossible. The
geodesic behaves very peculiar and is actually parrallel with itself after it circulates the
cone. This is a crucial idea because when we carry out the same process for generating
angles above 180 degrees we can see the line diverge from itself after it wraps around
the cone and extends to infinity never intersecting itself. This is our second classification of geodesics on a cone. Now that we have established what is a geodesic let
us form a classification system. We know that geodesics on cones with less than 180◦
intersect themselves infinitely many times.
Definition 4: Let the short path designate the portion of a geodesic that does not inter2
Figure 3: Geodesic on 90◦ Cone
Figure 4: Geodesic on 180◦ Cone
sect itself.
Definition 5: Let the Kth path designate the portion of a geodesic that intersects itself K times.
Definition 6: A geodesic exists between any two points in G2 .
Similar to helixes on a cylinder, non-vertical geodesics wrap around the cone infinitely
many times. Between any two points there exist infinitely many paths from the meager
short path all the way to the Nth path where N is approaching infinity.
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Figure 5: The Classification System
Triangles
We can create triangles by connecting three geodesics in various ways to produce triangles. In figure 6 we see the basic triangle that exists on all cones. In figure 7 we see
how to make a 135◦ Star Triangle on a 90◦ Cone. This shows that the lines used are
geodesics because they are straight on the plane. If you cut out the 90◦ cone and put
it together you see that each of the 3 straight lines, intersect themselves and connect to
two different points. Because it intersects, it produces this special shape which is why
we call it the star triangle.
Figure 6: Basic Triangle on a Cone using 3 Geodesics
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Figure 7: 135◦ Star Triangle on 90◦ Cone
Figure 8: Physical Model of 135◦ Star Triangle on 90◦ Cone
Circles
On G2 , the circle is defined in the same way that they are in E2 , that is to say the set
of points that are at the same distance along geodesic from the center. The first form
of circle looks like the one we have on the plane, except stretched to go around the
surface of the cone. The second form of circle is more interesting. If the center of our
cone is placed high enough and if the cone angle is small enough, and if the radius
is large enough, then we can have a circle that will completely wrap around the cone,
and intersect. This interesting form of circles can exist with repeated patterns along the
geodesic that spiral to the top of the cone.
The next shape of interesting circles is the one that have a radius that is greater
than the distance from the middle of the circle to the top of the cone. Such circles
have one of their points defined by the line that goes through the vertex as seen in
figure 9. Because of this, it is not obvious that the circle would be a continuous figure.
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Figure 9: Circle on a cone < 180◦
Experiments with this figure have showed us that the point that comes from the line that
goes through the vertex actually completes the rest of the points that will wrap around
near the tip of the cone, creating a circle that is continuous.
One of the interesting results we had with circles was that the circumference of the
circle who’s middle point (center of the circle) is at the vertex was affected not only by
the radius of the circle, but also by the angle of the cone. For a cone with angle 360, it
is the same as with E2 , but as the cone angle decreases, the circumference of the circle
deccreases and reaches 0 for θ = 0, and increases as the angle of the cones gets bigger
and bigger. As seen in Figure 10, the cone with π/2 angle will have by construction
circles with circumference 1/4 of 360 cones, and π will have 1/2 of the circumference
of 360 cones for the same radius, while a 450 cone will have 5/4π of the 360 cone. the
proposed formula for the circumference of a circle on a cone is for a cone of angle θ
and radius r:
Definition 7: The circumference of a circle, C, with radius r whose center lies on
the vertex of a cone of angle θ is given by:
C=
θ
∗r
2
Further Exploration
We noticed that there is a relation between two lines intersecting the vertex and a radial
circle as seen in figure 11 creating angles θ and γ. We did not have time to pursue this
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Figure 10: Development of Definition 7
but we believe there to be an equation relating the two.
Figure 11: Relation between γ and θ
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