The latter part of this equation
becomes 0
This result is in accordance to the theorem of Pappus
& Guldin: V=(2p
b)A
(Here we have b = 2 and
where r is the radius of the circle, r = 1)
The surface area of a Torus is found by multiplying
the circumference of the initial circle of
radius r and the circumference of the circle generated
by the center of mass of the initial circle around
the z-axis. That is:
Intersection of a torus and a plane
Cassinian ovals are the intersection of a torus and
a plane in certain position. Let b-a be the inner radius
of a torus whose generating circle has radius a. Cassinian
oval is the intersection of a plane parallel to the
torus' axis and a
distant from it. If b = 2 a, then it is the Lemniscate
of Bernoulli. Note that these tori in the figure are
not identical. Arbitrary slice of a torus are not Cassinian
ovals.
The Lemniscate of Bernoulli
Lemniscate of Bernoulli is a special case of Cassinian
ovals. That is, the locus of points P, such that distance[P,F1]
* distance[P,F2] = (distance[F1,F2]/2)^2, where F1,
F2 are fixed points called foci. It is analogous to
the definition of the ellipse, where sum of two distances
is replace by product.
We use a trigonometric substitution:
Thus equation (2) becomes:
The value of a
is in fact arctan(r/k)
b
is taken arbitrary here (it actually makes the plane
rotate around the z-axis but still with the same inclination
to the x-y plane)
Click on this to rotate the image
When the plane cuts the torus in this configuration,
i.e. when the plane cuts the torus
at its top point and bottom point on the surface of
the torus and through the center of the torus o, as
shown in the 3D graph above, the cross-section is in
fact two circles as we see in fig.5
Here, we find 2 circles located on the cutting plane
and on the surface of the torus. However, the plane
cuts the torus at all its height, the z-values going
from -r to r.
Let's say take a point P on one of the circle of fig.5.
This point P is fixed on the surface of the torus.
There will be then another point (let's call it P',
and P' is fixed on the plane) on the other circle that
has the same height (z coordinate) as P's. Now, imagine
you are rotating the plane, just the plane, around
the z-axis (you can do this by changing the value of
b)
. You'd see that the cross-section of fig5 is rotating
around its center. Rotate the plane until P' which
is on the cutting plane coincides with our fixed P
on the surface of the torus. So now our P lies on another
circle than the original one.
Conclusion: for any point P on the surface of any torus
which has genus 1, we can say that there are at least
four circles passing through P: one - the generating
circle with radius r.
two - the generated circle around
the z-axis with radius between k-r and k+r.
three - one of the circle generated
by cutting the torus with a plane tilted to the x-y
plane with an angle of arctan(r/k) and rotated around
the z-axis with a certain angle b.
four - one of the circle generated by cutting the torus
with the same tilted plane, but rotated around the
z-axis with the angle -b
.
Formulas
·
Parametric: Cos[t]/(1+Sin[t]^2) {1, Sin[t]}, 0 <
t <= 2 Pi.
·
Polar: r^2 == Cos[2 theta].
·
Cartesian: (x^2 + y^2)^2 == (x^2-y^2).
Foci are at {-1/Sqrt[2],0}, {1/Sqrt[2],0}
Slicing a Torus
The Lemniscate of Bernoulli is the intersection of a
plane tangent to the inner ring of a torus whose inner
radius equals to its radius of generating circle.
Cartesian equation:
After constructed conic sections by cutting a cone
by a plane, around 150 BC which was 200 years later,
the Greek mathematician Perseus investigated the curves
obtained by cutting a torus by a plane which is parallel
to the line through the centre of the hole of the torus.
In the formula of the curve given above the torus is
formed from a circle of radius a whose centre is rotated
along a circle of radius r. The value of c gives the
distance of the cutting plane from the centre of the
torus.
When c = 0 the curve consists of two circles of radius
a whose centres are at (r,0) and (-r,0).
If c = r+a the curve consists of one point, namely the
origin, while if c > r+a no point lies on the curve
A Mathematical Approach to Slicing Doughnuts
Figure 1 shows a torus, a surface studied by mathematicians
not just because it
reminds them of a chocolate doughnut or a bagel filled
with smoked salmon and
cream cheese, but also because it has useful applications
and interesting
mathematical properties. We are about to illustrate
just one such property.
The axes shown in red in the diagram just indicate
how we have placed the torus in the usual
coordinate system for three dimensional space.
Figure 1: A Garden Variety Torus
What we propose to do is slice the torus through with
a plane - much as you might cut a doughnut in half
with a large knife - and have a look at the "faces"
of the cut. However, instead of just cutting vertically,
we will try cutting at various different angles. The
only restriction is that the plane we slice with must
contain one of the horizontal axes shown in the diagram.
For the sake of convenience we will use the axis pointing
toward the viewer in the diagram.
Figure 2: The Simplest Slice
Figure 2 illustrates what happens if you slice the
torus in the way you would probably slice your doughnut
if you wanted to share it with a friend. Assuming you
wanted to share it fairly, you would just slice it
vertically throug the middle. It is easy to see (and
even check with a real doughnut) that the faces of
the cut are just a pair of circular disks.
Now, being mathematicians, we like to throw away all
of the unnecessary features and just deal with essentials.
So we assume that the torus has no interior. It's just
a hollow shell with no thickness. A doughnut like that
would leave you hungry, but mathematically speaking,
that's what a "torus" actually is. A torus
with its interior included is properly called a "solid
torus". Anyway, the "faces" of our vertical
cut are just circles now and we use the technical term
"intersection" to describe them.
Figure 2 also shows what how we intend to vary the
angle of the cut by rotating the plane about the axis
pointing toward the viewer.
Figure 3 shows the cut where the plane is horizontal.
This one may be a little difficult in practice if your
doughnut is soft, but it is easy to
see that the intersection consists of two concentric
circles.
Figure 3: Another
Easy Slice
So now let's watch what happens to the shape of the
cut as we rotate the plane
through 360 degrees. This is what is happenning in
figure 4 (fig.4 is an animated gif file, Open fig4.
with Netscape). Starting from the horizontal position
where there are two concentric circles we go through
the vertical position where there are two separate
circles and then back again. We are
viewing the plane of intersection at right angles
at every stage of the animation.
Observe that the two separate circles gradually deform
into kidney shapes and
then, somewhere along the line, they cross over into
the concentric shapes which
eventually become the concentric circles at the horizontal
position.
The four circles on the Torus:
Polar parametric equations for a plane:
And here is the cutaway view of the torus:
It is also possible to write the equation of the torus
as 3 parametric equations:
where
and
Changing the values of k & r
Name: Alexandre BOUQUIN
Dayne MATSUMOTO
you can actually play by changing the r and R (which
is k) values
to change the shape of the Torus.
Definition:
A surface in the shape of a doughnut with a hole in
it; a surface generated by the rotation , in space,
of a circle about an axis in its plane but not cutting
the circle. If
r
is the radius of the circle,
k
the distance from the center to the axis of revolution,
in this case, the z-axis, and the equations of the
generating circle is
, then the equation of the anchor ring is:
This is the cartesian form of the equation of the torus.
Try also: when k=0
and when r=0
when k=0, that means that the distance of the center
of the circle to the axis of rotation is 0.
So, we get
A torus is an object of genus1, that means it has one
"hole".
Definition
: the genus of a surface is the largest number of non-intersecting
simple closed curves that can be drawn on the surface
without separating it into two unconnected parts.
Theorem of Pappus and Guldin:
Let R be a plane region lying completely to one
side of a line l.
Then the volume V of the solid region generated
by revolving R
about l is given by
V=
(2
pb)A
where A is the area of R and b is the distance
from the center
of gravity of R to the line l.
Example: The circular
disk
is rotated around the y-axis. Find the volume of the
torus generated.
Solution: The
circular disk
consists of the circle centerd at (2;0) with radius
1 and all points interior into it.
In the upper part of the circle (that is, in the first
quadrant) y=
while in the lower part of the circle, y is negative
and is equal to -
. Thus the height of a typical cylindrical shell is
equal to:
Thus the volume V of the torus is (by cylindrical shells):
(1)
