The Intersection of Two Cones Sharing the Same Cross-section
Abstract
We know that conic section have many interesting property and it can
be generated by the intersection of a plane with one or two nappes of a
cone.
If we let a conic section be the cross-section of two different cones,
then we observe that the intersection of two cones will be a conic section, too.
In Chapter 1 and 2, we would state in detail how to use Cabri 3D to draw the figure of the intersection of two cones. In Chapter 3, we introduce the property of tangent lines on the two conic section.
Finally, we consider a special case in the figure.To evaluate the volume
of the intersection of two cylinders will be an interesting question in Calculus.
Chapter 1.
The Intersection of Two Cones
In the beginning, there are two non-degenerate cones determined by an ellipse on xy-plane and two different vertexs A and B on space.
Figure 1.1
Construct a plane passing through points A and B such that it has two intersection points C and D with the conic section on xy-plane.
Figure 1.2
Give straight lines ,, and , let point E be the intersection point of and , point F be the intersection point of and .
Figure 1.3
Construct a plane passing through A and B such that it has two intersection points G and H with the ellipse on xy-plane.
Figure 1.4
Give straight lines and , let point I be the intersection point of and .
Figure 1.5
Construct a plane J passing through I, E and F.
Figure 1.6
Let conic section K be the intersection of plane J and cone with vertex A.
Figure 1.7
Chapter 2.
The Intersection of Two Cones will on a Plane
Lemma
Let K be the conic section on a plane and there is a point A outside
the conic section K on the same plane. Let be a straight line passing through point A intersects K at different points B and C, be a straight line passing through point A intersects K at points D and E. Construct four straight lines , , and . Let point F be the intersection of and , G be the intersection of and . Then the straight line passing through points F and G is dependent only on the point A and the conic section K, i.e is fixed when , are moved.
Figure 2.1
See the figure 1.7 again.
Figure 2.2
Construct straight line , let intersects xy-plane at point O.
Figure 2.3
is the intersection of and , which is passing through points A, B, C, D and is passing through points A, B, G, H. So point O will be the intersection of , and xy-plane. Since and intersect xy-plane on and , respectively. It implies that point O will be the intersection of and .
Figure 2.4
Construct four straight lines , , and . Let point M be the intersection of , and N be the intersection of , . Then by Lemma, the straight line L passing points M, N is fixed.
Figure 2.5
Observed that point M be the intersection of and xy-plane, point N be the intersection of and xy-plane. So points M and N are on the plane J. It implies that L is on the plane J and xy-plane, i.e L is the intersection of plane J and xy-plane.
Figure 2.6
Then plane J can be determined by L and I. As the point C moves along the ellipse on xy-plane, the plane J is fixed but the point E moves along the conic section K. Because point E is on the intersection of two cones, the intersection of two cones is on the plane J. Hence the intersection of two cones will be the conic section K.
Chap 3.The property of the tangent lines on the intersection
Now we have two cones sharing the same ellipse and the intersection of them. Let plane L passing through points A and B contains M, N, O and P, which points M, N on the ellipse on xy-plane and points O, P on the conic section K.
Figure 3.1
Therefore the four tangent lines of conic section passing throught M, N, O and P will intersect at one point Q. Hence point Q will be on the line, which is the intersection of plane J and xy-plane.
Figure 3.2
We will introduce the way of finding the tangent line of ellipse on the Appendix A.
Chapter 4.
Application
Consider the special case which the vertex of cones are at infinity, it will become two cylinders and determined by straight lines , and the radius of circle r as the following figure.
Figure 4.1
First we construct two bisector planes and between straight lines and .
Figure 4.2
Let be the intersection of and , be the intersection of and .
Figure 4.3
Now we have conic section and .
Figure 4.4
Construct a plane perpendicular to the z-axis intersects at points A and C, intersects at points B and D.
Figure 4.5
Construct a parallelogram with vertexes points A, B, C and D.
Figure 4.6
Now we can evaluate the volume of intersection of two cylinders by
integral the parallelogram with respect to z from -r to r.
Figure 4.7
The detail of the process of calculation is left to the Appendix B.
Appendix A
Construct tangent line on an ellipse
Now we have an ellipse on plane and a point A on ellipse.
Figure A.1
Let A be a vertex, draw a pentagram which inscribed in the ellipse passing other points B, C, D and E.
Figure A.2
Let point F be the intersection point of and , point G be the intersection point of and .
Figure A.3
Let point H be the intersection point of and .
Figure A.4
Then be the tangent line of the ellipse at point A.
Figure A.5
Appendix B
Evaluate the Volume of The Intersection of Two Cylinders
Give the angle between straight line and be θ. Now we have a parallelogram with vertexes points A, B, C and D on the plane z = h, i.e the plane . Let the point (0,0,h) be E.
Figure B.1
First, we construct a circle with axis intersect z-axis at (0,0,r),
then we get a half of the height of the parallelogram is .
Figure B.2
Construct a segment , since is parallel to and is parallel to , the angle between and is θ. We get the angle between and which is θ/2. It Implies that the distance of point E and point D is
.
Figure B.3
Construct a segment . Since and are on the different bisector planes between and , the angle between and is a right angle. We will get a right triangle ADE. Hence the length of is
.
Figure B.4
Let the volume of the intersection of two cylinders is V. Then V is
,
,
and 
, let point E be the intersection point of 
and 
, let point I be the intersection point of 
be a straight line passing through point A intersects K at different points B and C, 
be a straight line passing through point A intersects K at points D and E. Construct four straight lines 
, 
, 
and 
. Let point F be the intersection of 
and 
, G be the intersection of 
and 
. Then the straight line 
passing through points F and G is dependent only on the point A and the conic section K, i.e 
is fixed when 
, 
are moved.
, let 
intersects xy-plane at point O.
is the intersection of 
and 
, which 
is passing through points A, B, C, D and 
is passing through points A, B, G, H. So point O will be the intersection of 
, 
and xy-plane. Since 
and 
and 
, respectively. It implies that point O will be the intersection of 
and 
. 
, 
, 
and 
. Let point M be the intersection of 
, 
and N be the intersection of 
, 
. Then by Lemma, the straight line L passing points M, N is fixed. 
and xy-plane, point N be the intersection of 
and xy-plane. So points M and N are on the plane J. It implies that L is on the plane J and xy-plane, i.e L is the intersection of plane J and xy-plane. 
and 
determined by straight lines 
, 
and the radius of circle r as the following figure.
and 
between straight lines 
be the intersection of 
be the intersection of 
and 
, point G be the intersection point of 
and 
.
and 
.
be the tangent line of the ellipse at point A.
and 
be θ. Now we have a parallelogram with vertexes points A, B, C and D on the plane z = h, i.e the plane 
. Let the point (0,0,h) be E.
intersect z-axis at (0,0,r), 
.
, since 
is parallel to 
and 
is parallel to 
and 
is θ. We get the angle between 
and 
which is θ/2. It Implies that the distance of point E and point D is 
.
. Since 
and 
are on the different bisector planes between 
and 
, the angle between 
and 
is a right angle. We will get a right triangle ADE. Hence the length of 
is 
.
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