In the article "Why a Parabola? — A Study of Reflective Surfaces", we derived from optical properties that the curve satisfying those properties is a parabola, and at the same time, we couldn’t help but feel the beauty of vector analysis. Perhaps some readers will feel there is more to explore: there are three types of conic sections, but the article only introduced one. Well, in this article, we will start from two other optical properties and derive the curves (ellipse and hyperbola) that satisfy them.
(Note: In the following descriptions, bold vectors represent light rays, and the curves represent reflective surfaces.)
I. Light rays emitted from one point converge at another point after being reflected by a curve (surface).
This is a property of an ellipse, and we want to prove that only an ellipse satisfies it. As shown in the figure above, let the vector \vec{r} (AC) represent the trajectory of the curve, and let \vec{F} represent the distance vector between A and B (this is a constant vector). Let \vec{R} = \vec{F} - \vec{r}, and denote |\vec{r}| = r and |\vec{R}| = R (unless otherwise stated, Scientific Space adopts these two notations).
According to the law of reflection, the angles between vectors \vec{R} and \vec{r} with respect to CE (d\vec{r}) are equal, so we have:
\frac{\vec{r} \cdot d\vec{r}}{r} = \frac{\vec{R} \cdot d\vec{r}}{R}
Don’t forget the relationship \vec{R} = \vec{F} - \vec{r}. We have d\vec{r} = -d(\vec{F} - \vec{r}) = -d\vec{R}. Substituting this into the above equation, we get:
\frac{\vec{r} \cdot d\vec{r}}{r} + \frac{\vec{R} \cdot d\vec{R}}{R} = 0
We return to the identity \vec{r} \cdot d\vec{r} = r dr. The above equation is equivalent to dr + dR = 0. Integrating both sides gives:
r + R = C \Rightarrow |\vec{r}| + |\vec{F} - \vec{r}| = C
Is this not the definition of an ellipse? Thus, the problem is solved. Next, let’s move on to the next task...
II. Light rays emitted from one point reflect as if they were emitted from another point.
With the experience gained from analyzing the optical properties of both the ellipse and the parabola, I believe readers have accumulated some methods. Therefore, I will not repeat the descriptive derivation process and will write directly:
\begin{aligned} \vec{R} &= \vec{r} - \vec{F} \\ \frac{\vec{r} \cdot d\vec{r}}{r} &= \frac{\vec{R} \cdot d\vec{r}}{R} \end{aligned}
Since d\vec{r} = d(\vec{r} - \vec{F}) = d\vec{R}, substituting this into the equation yields:
\frac{\vec{r} \cdot d\vec{r}}{r} - \frac{\vec{R} \cdot d\vec{R}}{R} = 0
which is equivalent to:
dr - dR = 0
Integrating both sides gives:
r - R = C \Rightarrow |\vec{r}| - |\vec{R}| = C
Similarly, this is the definition of a hyperbola! Thus, the discussion is concluded. Once again, we have concisely solved the problem that troubled us using vectors!
Finally, here is some history regarding conic sections (collected from the web):
Historically, the first to examine conic sections was Menaechmus (375 BC–325 BC); about 100 years later, Apollonius studied conic sections more thoroughly and systematically. Their study of conic sections was very practical: examining the curves obtained by cutting a cone with planes of different inclination angles. That is, if the angle between the cut and the base is smaller than the angle between the generator and the base, the cut appears as an ellipse; if the two angles are equal, the cut appears as a parabola; if the former is greater than the latter, the cut appears as a hyperbola. Furthermore, Apollonius further studied the optical properties of these conic sections. For example, with an ellipse, he discovered that if one side of the focal point F is made into a mirror and a light source is placed at F, all light reflected by the elliptical mirror passes through the other focal point F'. Since heat reflects just like light, it would be scorched at that point, which is the origin of the name "focus" (focal point). It is said that he made this discovery while studying the construction of an ellipse (the method introduced at the beginning of current textbooks).
The true transition of conic sections from the background to the foreground, from the academic ivory tower into the world of real life, should be credited to the German astronomer Johannes Kepler (1571 AD–1630 AD). Through long-term astronomical observations and data analysis, Kepler discovered the famous "Kepler’s Three Laws," the first of which is: "Planets move in a plane containing the Sun, tracing an ellipse with the Sun at one focus." In this way, the curves studied by Menaechmus and Apollonius out of mathematical interest debuted on the stage of astronomy nearly 2,000 years later. Later, Halley used the theory and calculation methods of conic sections to accurately predict the moment when Halley’s Comet would be closest to the Earth. In 1758, 16 years after Halley’s death, Halley’s Comet met the Earth as scheduled, causing a sensation throughout Europe and the world, and further promoting people’s interest in the study of conic sections.