Circular cylinder, plotted using MathCad.
In this post I explore some of the interesting aspects of surfaces and their analytic geometry What we will find is that in many respects, there are similarities to plane analytic geometry, e.g.
http://brane-space.blogspot.com/2017/01/seen-hidden-figures-heres-look-at.html
including the use of analytical equations to examine the properties or what will now be solids or 3D surfaces, as opposed to plane figures like circles, ellipses, parabolas etc. In general, given some locus of points P(x, y, z) satisfying an equation:
F(x,y, z) = 0
One will generate a surface. The simplest examples are planes, yielding linear equations involving only first powers of the variables,, e.g. x + y + z = 0 . The next simplest surfaces are the cylinders for which: F(x,y) = 0 and the x, y are raised to the 2nd power. In gneral, a cylinder is a surface generated by a straight line that moves parallel to a given line and passes through a given curve. For example, the given curve may be:
x 2 + y 2 = r 2
One of the first things the math enthusiast needs to be aware of is that the equations need to be seen in context. In the case of the cylinder surface, such as shown in the top graphic, the cross-sectional area is a circle and also the given curve, i.e. in the x-y plane. Here the generator of the cylinder is also parallel to the z-axis, so z = 0, and therefore z would not be included in the equation.
On inspection, we see the radius r of the cross section area circle is 5 units on either side. Hence, the applicable equation for this cylinder will be:
x 2 + y 2 = (5) 2
Now consider this equation, also for a cylinder:
y = x 2
At first glance this may be recognized as a parabola, but in the context it is really a parabolic cylinder, i.e. having parabolic cross section and in the plane z = 0. Below I show the detailed image for the cylinder and adjacent to it the MathCad plot.
Again, note carefully that the given equation from which one letter is missing - in Cartesian coordinates- represents in space a cylinder with elements parallel to the axis associated with the missing letter. In a similar vein, which the math enthusiast can check, the surface with equation: y 2 + 4 z 2 = 4
represents an elliptic cylinder with elements parallel to the x- axis. It extends indefinitely in both positive and negative directions along the x-axis which is now called the axis of the cylinder, e.g.
In this case the cross -section area is that of an ellipse, which the reader can check by graphing.
All cylinders can be analyzed using analytical equations similar to the one we employed in plane analytic geometry, i.e.
Ax 2 + B xy + C y 2 + Dx + Ey + F = 0
(With elements taken parallel to the z -axis hence all equations in terms of x and y)
The Ellipsoid & Sphere:
Consider the ellipsoid shown above, with the origin of the coordinate axes (x,y,z) at its center. We take a,b,c to be positive constants and the relevant general equation here is:
x2 / a 2 + y2 / b2 + z2 / c2 = 1
Note the ellipsoid cuts coordinate axes at: (+ a, 0, 0), (0, +b, 0) and (0, 0, +c). Note all the sections cuts out by the coordinate axes are all ellipses, e.g.
x2 / a 2 + y2 / b2 = 1
And further when we have a = b = c the sphere is generated. For the special case of the sphere which does not have center at the origin of coordinate (e.g. x, y, z) we can write:
Clearly, just as we have plane curves which we recognize as circles, ellipses and parabolas, we have surfaces which we see are spheres, ellipsoids and paraboloids.
Consider now the elliptic paraboloid below, for which: x2 / a 2 + y2 / b2 = z/ c
And for which I have also provided the corresponding MathCad plot. Note the surface is symmetrical with respect to the planes z = 0, and y= 0, the only intercept on the axes is at the origin (not as clearly shown in the MathCad plot because of limitations in the program) The key point is that the section cut out from the surface by the yz-plane us:
z = 0, y2 = (b2 / x ) z
This is a parabola with vertex at the origin and opening upward. Similarly, when:
y = 0, x 2 = (a2 / c ) z
One also has a parabola. Note that when z = 0 the cut reduces to the single point: (0, 0, 0)
We now examine the elliptic cone given by the equation:
x2 / a 2 + y2 / b2 = z2 / c2
Which object is symmetrical with respect to all three coordinate planes, i.e.
(Note: The cone is also below the x-y plane shown as a 'mirror image' of the cone above the plane. It's just that the MathCad program is limited in showing the dual aspect)
The plane z = 0 cuts the surface n the single point (0, 0, 0). In addition, the plane x = 0 cuts it in the two intersecting straight lines:
1) x = 0, y/b + z/ c
2) y = 0, x/ a = + z/ c
The section cut out by the plane z = z1 > 0 is an ellipse with center on the z-axis and vertices on the straight lines given by the preceding eqns.
Example Problem:
Describe and sketch the surface given by: r = 2 cos Θ
The characteristics of the equation indicate the surface is a right circular cylinder, e.g.
The given coordinate system is cylindrical, e.g. r, Θ , z. Sketching the equation yields a circular cross section of radius 1 unit. The cylinder- with the displayed cross section area- extends indefinitely in both the positive and negative directions along the z axis which intersects at (1,0) in the rectangular frame shown. The polar view is shown to the right.
Suggested Problems:
1) A surface configuration is described by the following analytical aspects:
A= 1, B = 1, C = 2, D = 1, E = 3, F = 1
Write the equation for the object and identify it.
2) Describe and sketch each of the following:
a) x 2 + y 2 = a 2
b) x 2 + y 2 + z 2 + 4x - 6y = 3
3) Sketch (and identify) the surface given by:
f(x,y) = 4 - x 2 + y 2
represents an elliptic cylinder with elements parallel to the x- axis. It extends indefinitely in both positive and negative directions along the x-axis which is now called the axis of the cylinder, e.g.
In this case the cross -section area is that of an ellipse, which the reader can check by graphing.
All cylinders can be analyzed using analytical equations similar to the one we employed in plane analytic geometry, i.e.
Ax 2 + Ay 2 + Dx + Ey + F = 0 (A not equal 0)
In this case we will use for various cylinders:
Ax 2 + B xy + C y 2 + Dx + Ey + F = 0
(With elements taken parallel to the z -axis hence all equations in terms of x and y)
The Ellipsoid & Sphere:
Consider the ellipsoid shown above, with the origin of the coordinate axes (x,y,z) at its center. We take a,b,c to be positive constants and the relevant general equation here is:
x2 / a 2 + y2 / b2 + z2 / c2 = 1
Note the ellipsoid cuts coordinate axes at: (+ a, 0, 0), (0, +b, 0) and (0, 0, +c). Note all the sections cuts out by the coordinate axes are all ellipses, e.g.
x2 / a 2 + y2 / b2 = 1
And further when we have a = b = c the sphere is generated. For the special case of the sphere which does not have center at the origin of coordinate (e.g. x, y, z) we can write:
(x - h) 2 + (y - k ) 2 + (z - m ) 2 = a 2
Clearly, just as we have plane curves which we recognize as circles, ellipses and parabolas, we have surfaces which we see are spheres, ellipsoids and paraboloids.
Consider now the elliptic paraboloid below, for which: x2 / a 2 + y2 / b2 = z/ c
And for which I have also provided the corresponding MathCad plot. Note the surface is symmetrical with respect to the planes z = 0, and y= 0, the only intercept on the axes is at the origin (not as clearly shown in the MathCad plot because of limitations in the program) The key point is that the section cut out from the surface by the yz-plane us:
z = 0, y2 = (b2 / x ) z
This is a parabola with vertex at the origin and opening upward. Similarly, when:
y = 0, x 2 = (a2 / c ) z
One also has a parabola. Note that when z = 0 the cut reduces to the single point: (0, 0, 0)
We now examine the elliptic cone given by the equation:
x2 / a 2 + y2 / b2 = z2 / c2
Which object is symmetrical with respect to all three coordinate planes, i.e.
(Note: The cone is also below the x-y plane shown as a 'mirror image' of the cone above the plane. It's just that the MathCad program is limited in showing the dual aspect)
The plane z = 0 cuts the surface n the single point (0, 0, 0). In addition, the plane x = 0 cuts it in the two intersecting straight lines:
1) x = 0, y/b + z/ c
2) y = 0, x/ a = + z/ c
The section cut out by the plane z = z1 > 0 is an ellipse with center on the z-axis and vertices on the straight lines given by the preceding eqns.
Example Problem:
Describe and sketch the surface given by: r = 2 cos Θ
The characteristics of the equation indicate the surface is a right circular cylinder, e.g.
The given coordinate system is cylindrical, e.g. r, Θ , z. Sketching the equation yields a circular cross section of radius 1 unit. The cylinder- with the displayed cross section area- extends indefinitely in both the positive and negative directions along the z axis which intersects at (1,0) in the rectangular frame shown. The polar view is shown to the right.
Suggested Problems:
1) A surface configuration is described by the following analytical aspects:
A= 1, B = 1, C = 2, D = 1, E = 3, F = 1
Write the equation for the object and identify it.
2) Describe and sketch each of the following:
a) x 2 + y 2 = a 2
b) x 2 + y 2 + z 2 + 4x - 6y = 3
3) Sketch (and identify) the surface given by:
f(x,y) = 4 - x 2 + y 2
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