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)) x^{2}. 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)) x^{2}. 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)) x^{2}. So,
Also at this point, the horizontal displacement from the axis of revolution is x:
In the circular cross-section of the paraboloid, therefore, 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:
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.
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)) x^{2}. 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)) x^{2}. 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)) x^{2}. So,
Also at this point, the horizontal displacement from the axis of revolution is x:
In the circular cross-section of the paraboloid, therefore, 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:
y_{e} | y_{e} | |
-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.
Labels: conic section, ellipse, elliptical section, solid of revolution