Plane Through Parabolic Solid of Revolution

Several previous posts examined properties of the curves produced when a cutting plane passes through a right cone at an angle (to produce a section whose edge describes a parabola or ellipse.)

A right cone can be thought of as a Solid of Revolution (SOR), produced by revolving the appropriate straight line around an axis. For example, the following image shows a portion of the graph for the equation y = 100 - 2*x revolved around the y-axis:

When this line is revolved around the y-axis, the familiar right cone is produced, through which a cutting plane can be passed to produce one of the familiar conics (parabola, circle, ellipse, and hyperbola.)

I would now like to generalize this examination by extending the concept to SORs produced by revolving non- straight lines around the y-axis. In other words, I will examine the shapes produced when a cutting plane passes through a body whose body has a circular cross-section perpendicular to the axis of revolution--but is not a right cone.

For the present example, an SOR is examined that has been produced by revolving a parabolic line around the y-axis. The line is described by
y = [1/(4c)] x^2, where c = 0.143823832162.

The SOR produced is shown in the following image (it may appear upside down compared to other images, but the orientation doesn't matter; the same shape is produced, which is all that is important for the sake of this investigation):

Passing a cutting plane through it,

And with the top section removed to see the resulting plane:

Looking at a cross-section (through the mid-point) of the paraboloid from the side, note a few basic features:

The origin of the coordinate system of the 2-D parabola that was used to create the paraboloid coincides with the vertex of the parabola:
positive X is to the right;
positive Y is up, and also acts as the axis of revolution, becoming the axis of the paraboloid; and
positive Z is coming out of the page toward the reader.

α is the angle at which the cutting plane goes through the paraboloid, with respect to the horizontal.

h_c is the vertical distance from the vertex of the parabola to the point where the cutting plane intersects the axis of revolution. At this point, r_c is the radius of the paraboloid.

r_u is the distance from the axis of revolution to the upper edge of the paraboloid, measured along the inclined plane.
r_l is the distance from the axis of revolution to the lower edge of the paraboloid, measured along the inclined plane.

Affix a new coordinate system to the plane (x_e, y_e, z_e). Looking at the section from above, and perpendicular to, the plane of the section, it appears as illustrated in the following image:

Positive x_e is along the plane to the right, measured from the mid-point of the plane;
positive y_e is up along the plane, measured from the mid-point of the plane; and
positive z_e is perpendicular to the plane, coming out of the page toward the reader.

Now to calculate some of these parameters.

r_l must satisfy two conditions (imposed by the original parabola):
r_l cos(α) = x     AND     h_c - r_l sin(α) = y = (1/(4c)) x². Solving these equations yields

Similarly, r_u must satisfy two conditions:
r_u cos(α) = x     AND     h_c + r_u sin(α) = y = (1/(4c)) x². Solving these equations yields

If the entire length across the middle of the plane is 2a = r_u + r_l, then half the length is a = (r_u + r_l)/2:

This mid-point of the plane is offset from the axis of revolution by a distance r_o measured along the plane. This parameter can be found either as r_u - a or a - r_l; they yield the same result:

Note that r_o is independent of h_c; it is a function of c and α only.

If the condition that a = 1 is imposed, expressions for c and h_c in terms of each other can be derived, as well as a modified expression for r_o:

The coordinate system on the cutting plane will now be used to compute some points.
Say we go along the plane a distance x_e from the mid-point of the plane. At this height, the circular cross-section of the paraboloid will have a radius, r_xe.
This point must satisfy the equation h_c + (r_o + x_e) sin(α) = y = (1/(4c)) x². So,

Also at this point, the horizontal displacement from the axis of revolution is x:

In the circular cross-section of the paraboloid this point represents an angle of

Finally, the y_e along the plane is found as

Now to create a numeric example to plot the shape of the plane. Say
h_c = 0.431235852853
α = 0.904556894307 radians.
Then,
a = 1
r_o = 0.592028088783

Computing the y_e values for several x_e values, the following table is produced:

xe		ye
-1		0
-0.9		0.269394770065
-0.8		0.370820393251
-0.7		0.441364549706
-0.6		0.494427190998
-0.5		0.535233134658
-0.4		0.566437507098
-0.3		0.589566849675
-0.2		0.60554716645
-0.1		0.614936054508
0.0		0.618033988748
0.1		0.614936054509
0.2		0.60554716645
0.3		0.589566849674
0.4		0.5664375071
0.5		0.53523313466
0.6		0.494427191002
0.7		0.441364549705
0.8		0.370820393255
0.9		0.269394770066
1		0

A plot of this data indicates that the plane is an ellipse. In fact, plugging each of the data points into the standard equation for an ellipse

where b is taken to be the mid-point of the section, confirms that the cutting plane is, indeed, an ellipse. The b point is found as

and the eccentricity of the ellipse is

Note that e is independent of h_c and c; it is a function of α only.
In the present example, the eccentricity, e, of the ellipse is e = 0.786151377767.
 
 

Conical Sections: A Deeper Look Part 2

This post continues from the previous post.
The previous post presented some very fundamental features of a cone and the ellipse that is created by a cutting plane going through it at an angle. I would now like to pick a particular cone and cutting angle to use for a numerical example. At this point, any cone-angle combination is equally suitable. However, perhaps by examining some other features of a conic section, a particular section will suggest itself for use as a numerical example.

Some conditions:

First, the cutting angle, α, must satisfy the following condition: 0 < α < arctan(γ).
If the angle is too big or too small, the resulting section will not be an ellipse; it will be a circle, parabola, or hyperbola.

Second, at the bottom of the previous post, I determined that the value of α for which r_u is a minimum is α = arctan(1/γ).
So, I will restrict my examination to right cones for which γ > 1.

Third, because I am hoping this exploration will reveal some information that may provide further insight applicable to elliptical motion, I am going to impose one more condition:
a circular section going through the same right cone, of equal area of the ellipse, must be at the same height on the cone as the mid-point of the ellipse.
Area of a circle: π r ²
Area of an ellipse: π ab
Therefore, h_qa = γ √(ab) = h_c + r_o sin(α)
After some algebraic manipulation, this reduces to
cos(α) = (1 - e²)^(1/4)
Which, in turn, leads to cos(α) = (1/γ).

So now a condition exists to define a particular class of elliptical section, which I am calling a qa-section (an elliptical section produced such that its equal-area circle is at the mid-point of the ellipse). Following are some properties of these qa-sections (including parameters from my previous post which can be simplified under these conditions).

Summary of Features for a qa Elliptical Section

However, even after applying these restrictions, an infinite number of sections is still possible. Perhaps a graph will illuminate some feature I can use. On the following image are two graphs: α_qa and α_rumin versus γ.
α_qa (the red line) is α for a qa-section (= arccos(1/γ)).
α_rumin (the blue line) is the value of α for which r_u is a minimum.

These lines intersect at only one point; this point suggests the cone-cutting plane I will use for numerical examples.
In fact, this point of intersection occurs when arccos(1/γ) = arctan(1/γ)
γ is then found as the solution to the following equation: γ ⁴ - γ ² - 1 = 0
This equation has four solutions:

These solutions correspond to γ = +/- 1.272019649514069 and +/- 0.786151377757i.

The imaginary roots and the negative real root are meaningless in the present context; only the real, positive, solution is applicable:
γ = 1.272019649514069

Alternately, these roots could have been found by forming the Companion Matrix for this polynomial and computing its eigenvalues and eigenvectors. The eigenvalues are the roots of the original polynomial. The Companion Matrix is

Two eigenvalues are +/- 0.7861513777574233i, and the associated eigenvectors are

The other two eigenvalues are +/- 1.272019649514069, and the associated eigenvectors are

So the roots have been confirmed.
This cone (γ = 1.272019649514069), and all the properties that follow from it, will be used for numerical examples going forward.

Some Numbers for the qa Elliptical Section for γ = 1.272019649514069

(a = 1)

γ = 1.272019649514069
α = 0.666239432488 radians
e = 0.786151377754
r_u = 0.381966011254
r_l = 1.61803398875
r_o = 0.618033988738
r_c = 0.485868271774
h_c = 0.618033988764
b = 0.618033988764
h_qa = 1
r_qa = 0.786151377767
cos(α) = 0.78615137776
sin(α) = 0.618033988746 = cos²(α)
tan(α) = 0.78615137776

Note that the expression under the radical returns itself:
0.618033988738 = √(1 - √ (1 - e²))
0.618033988738 = √(1 - √(1 - √ (1 - e²)))
etc.
This could go on forever.
In other words, this number satisfies the infinite expression
x = √(1 - √ (1 - √ (1 - √ (...))))

Before closing this post, I would like to point out that simply projecting a circle onto a plane inclined at an angle of 0.666239432488 radians would not produce an ellipse of eccentricity e = 0.786151377754. In fact, you would produce an ellipse, but not one of this eccentricity.
To confirm this assertion, y-coordinates of a circle have been multiplied by cos(α) and tabulated in a table in a previous post (Blog Post: Ellipse Sample Datapoints), under the column labeled y_con. Note that the values tabulated in this column do not product an ellipse of eccentricity 0.786151377754. In fact, an ellipse of eccentricity 0.618033988747 is produced. To produce an ellipse of the same eccentricity as the one produced by the conic section--and a cutting plane of the same angle, α--the y_con values have to be multiplied by an additional factor of cos(α).
 

Conical Sections: A Deeper Look Part 1

This post goes into more details about elliptical sections that are created by taking an inclined cutting plane through a right cone. This topic was briefly investigated in a previous post, Blog Post: Ellipse by Conical Section; here I investigate more deeply the details of ellipses by conical sections.

A 3D image of a cone and cutting plane are presented in the post mentioned above. Just below is an image of a side view of the cut going through the cone:

In this image, several variables are measured with respect to the axis of the cone.
H is the overall height of the cone.
R is the radius of the bottom of the cone.
These two characteristics of the cone are used to define another parameter, the height-to-radius ratio: γ = H/B

α is the angle at which the cutting plane goes through the cone, with respect to the horizontal.

h_c is the vertical distance from the tip of the cone to the point where the cutting plane intersects the axis of the cone.
r_c is the radius of the cone where the cutting plane intersects the axis of the cone (= h_c/γ).

r_u is the distance from the cone axis to the upper edge of the cone, measured along the inclined plane.
r_l is the distance from the cone axis to the lower edge of the cone, measured along the inclined plane.

Looking more closely at one half of the cross-section, a few more properties of the section can be introduced, as illustrated in the following image:

r_o is the distance from the cone axis to the center of the ellipse, measured along the elliptical plane.
h_qa is the vertical distance from the tip of the cone to the horizontal plane on which lies the center of the ellipse.
r_qa is the horizontal distance from the cone axis to the center of the ellipse.

At this point, enough parameters have been introduced. Now it is time to start defining them.

or

or

a, the length of the semi-major axis of the ellipse, can be found as follows:

r_u + r_l = 2a

Therefore,

b, the length of the semi-minor axis of the ellipse, is

Some generic relationships between the basic parameters of the cone and the cutting plane can also be defined:

Just out of curiousity, I would like to confirm the value of α for which r_u is a minimum. Differentiating the expression for r_u above with respect to α, and setting the resulting equation equal to 0, we find that r_u is a minimum when α = arctan(1/γ) = π/2 - arctan(γ).
In other words, r_u is a minimum when the cutting plane goes through the cone at an angle that is perpendicular to the side of the cone.
A similar examination using r_l leads to the same requirement.