What is that light at the bottom of my mug?
Figure 1
The play of light at the bottom of a mug is a phenomenon that has captured my attention time
and again. It’s not just about the presence of light, but rather the distinct and captivating shape it
assumes. Instead of taking the form of an ordinary and uninspiring point of light, it takes a
fascinating turn. It transforms into a perfect curve that resembles a heart, or a cardioid. In my
opinion, this is a small, yet intriguing detail in my everyday life, and one that I still have no
explanation for.
Background and Analysis
In order to answer our question and discover the reason behind this light, we need to explore the
topic of envelopes and how they relate to families of curves. To begin, we can represent a curve
implicitly as a function of x and y, where f(x,y) = 0. For instance, a line can be represented by x
2
+ y = 0, and a parabola by 𝑦 − 𝑥 = 0.
A family of curves refers to a group of related curves, each described by a parameter t. For
example, a curve in the family might be represented by t = 0, another by t = 1.5, and yet another
by t = 3.14.
We can create an entire family of curves by adding a parameter t to the implicit function. For
example, f(x,y,t) = x + y + t = 0 would produce a set of lines. Each value of t would generate a
different line in the family, causing the set to shift up and down accordingly.
Now, let's delve into the concept of envelopes. The envelope of a family of curves is a curve that
is tangent to every curve in the family at each point of contact.
To create a new curve from a family of related curves, we can use the same method as before. In
this case, the family of curves is represented by the lines that we have drawn. We start with one
line and then, as we vary the parameter "t", we shift to the next line in the family. While we have
only drawn a finite number of lines, if we were to consider all infinitely many lines in the family,
we would observe a perfect curve emerge. This curve is known as the envelope of the set of
curves.
Figure 2
Envelopes are fascinating because they provide a tangible representation of the underlying
mathematical concepts. The curvy part we see is the envelope, which is the curve tangent to
every member of the family of curves at every point.
However, it's important to note that not every family of curves has an envelope, and a family of
curves doesn't have to be made out of lines. They can be composed of wavy shapes or any other
type of curve. In some cases, there may even be multiple envelopes for a single set of curves.
(Figure 2)
To understand the mathematics behind envelopes, we use the concept of parameterization. Each
curve in the family is described by a parameter, which allows us to generate an entire set of
related curves. By adding a parameter to our implicit function, we can create a family of curves,
and each curve is represented by a unique value of the parameter.
The envelope of the family is then produced by taking the limit as the parameter approaches
infinity. This produces a curve that is tangent to every member of the family at every point. In
other words, the envelope is the curve that encompasses the entire family of curves.
To find the point on the envelope, we can examine two neighboring curves from the family with
similar parameters t, such as 0.5 and 0.51. For simplicity, let's assume that the curves are
represented by the implicit function f(x, y, t) = 0.
Figure 3
When we move from t = 0.5 to t = 0.51, the curve shifts slightly, and we can represent this shift
as a vector in the x-y plane. This vector is given by the partial derivatives of f with respect to x,
y, and t, evaluated at the point on the curve:
(dx, dy, dt) = (∂f/∂x, ∂f/∂y, ∂f/∂t)
We can think of this vector as pointing from the point (x, y) on the curve to the point (x + dx, y +
dy) on the neighboring curve.
Now, imagine that we start at the point (x, y) on the original curve, and we move a small distance
along the curve in the direction of this vector. This will take us to a new point on the neighboring
curve, which we can represent as (x + dx', y + dy').
The key idea is that as we make the parameter t vary continuously, the point (x + dx', y + dy')
traces out a curve in the x-y plane, which is tangent to the neighboring curve at (x, y). This curve
is the envelope that we are looking for.
To find the equation of this curve, we need to eliminate the parameter t. One way to do this is to
use the implicit function theorem, which tells us that if the partial derivatives of f with respect to
x and y are both nonzero at a point (x, y, t), then there exists a neighborhood of that point where
we can solve for x and y as functions of t.
In other words, if ∂f/∂x and ∂f/∂y are both nonzero at the point (x, y, t) on the curve, then we can
solve the equations f(x, y, t) = 0 and ∂f/∂x dx + ∂f/∂y dy + ∂f/∂t dt = 0 for dx/dt and dy/dt as
functions of x, y, and t. Then we can integrate these functions to find the equation of the
envelope curve in terms of x and y.
When considering a family of curves with a parameter t, there may exist a point on the envelope,
which is itself a smooth function, that lies on one of the curves in the family. By selecting two
neighboring curves with similar values of t, we can find the intersection point between them. As
the difference in their parameters becomes smaller, the intersection point will converge to a point
on the envelope. We can repeat this process for any value of t to find multiple points on the
envelope.
To mathematically define the envelope, we can use calculus. Specifically, we can consider a
point (x,y,t) on a curve in the family and determine the partial derivative of the function f with
respect to t, while holding x and y constant. If the partial derivative is zero, then small changes in
t do not result in any changes in x and y. This means that infinitesimally adjacent curves are
intersecting at the point (x,y,t) and therefore, it is a point on the envelope.
To fully describe the envelope, we need to satisfy two conditions. First, the point (x,y,t) must be
on a member of the family of curves, which is represented by the equation f(x,y,t) = 0. Second,
the partial derivative of f with respect to t must be zero at this point, as we just derived. By
finding all points that satisfy these conditions, we can define and calculate the envelope.
Application
To find the equation of the envelope of curves for the light at the bottom of the mug, we first
need to have a family of curves that describes the path of the light as it reflects off the mug.
Let's assume that the mug is a smooth, cylindrical surface with a circular base. If we shine a light
from above the mug, it will reflect off the surface at different angles depending on the position of
the light source.
To simplify the problem, let's consider only the case where the light source is directly above the
center of the circular base of the mug. In this case, the reflection of the light will form a family of
circles on the base of the mug.
Now, let's define a coordinate system with the origin at the center of the circular base of the mug.
We can describe the path of the light as it reflects off the mug using the equation of a circle:
(x - a)^2 + (y - b)^2 = r^2
where (a, b) is the center of the circle and r is its radius.
The position of the light source above the mug can be parameterized by an angle θ, with 0 ≤ θ <
2π. Let's assume that the distance between the light source and the base of the mug is fixed at
some value d.
Then, we can describe the path of the light as it reflects off the mug using the family of curves:
(x - dcos(θ))^2 + (y - dsin(θ))^2 = r^2
where r is the radius of the circle formed by the reflection of the light, and θ and d are parameters
that describe the position of the light source.
To find the envelope of these curves, we need to find the values of x, y, and r that satisfy both the
equation of the circle and its partial derivative with respect to θ:
(x - dcos(θ))^2 + (y - dsin(θ))^2 - r^2 = 0
-2d(x - dcos(θ))sin(θ) + 2d*(y - d*sin(θ))cos(θ) = 0
We can eliminate θ from these equations by solving for cos(θ) and sin(θ) and substituting the
results into the second equation. After some algebraic manipulation, we obtain the equation for
the envelope of the curves:
x^2 + y^2 - 2sqrt(x^2 + y^2) = r^2
This equation describes a hyperbolic curve with its focus at the origin and its asymptotes at x =
±d. The envelope of the family of circles formed by the reflection of the light on the base of the
mug is this hyperbolic curve.
Conclusion
In wrapping up, the seemingly ordinary phenomenon of light patterns at the bottom of a coffee
mug reveals a mathematical art when studied closely. The heart-shaped curve we observe is an
outcome of the principle of envelopes. Envelopes, essentially, are curves that barely touch every
curve within a family of curves at each corresponding contact point.
This enchanting illumination pattern is a physical manifestation of an envelope created by a
family of light rays being reflected at the mug's base. Envelopes, while being an abstract
mathematical concept, transpire in numerous shapes and forms - they need not be restricted to
straight lines but can take on the persona of a multitude of curves.
The envelope emerges from a family of curves as the parameter t stretches towards infinity. The
core component of discovering this envelope equation lies in a thorough analysis of the partial
derivatives of our implicit function that defines our family of curves.
References
Bruce, J. W.; Giblin, P. J. (1984), Curves and Singularities, Cambridge University Press
MathCurve. (n.d.). Enveloppe d’une famille de courbes, from
https://mathcurve.com/courbes2d.gb/enveloppe/enveloppe.shtml
Math Stack Exchange, (2018) Getting the envelope of a family of curves.
https://math.stackexchange.com/questions/2475863/getting-the-envelope-of-a-family-of-curves