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.