Caustic curves and dual curves

[mdoni]

Hello,
I'm investigating duality for plane curves, and I came across an 'original' interpretation of the Biduality theorem , that uses the notion of caustic curve. Because everything is still very obscure to me, I try to share the whole with you, in the hope that we can help to fix ideas.
Meanwhile, some definitions to introduce the argument:

Definition: Let $$\mathbb{P}^*$$ the projective plane dual of $$\mathbb{P}^2 (\mathbb{C})$$: every line of $$\mathbb{P}^2 (\mathbb{C})$$ identifies a point of $$\mathbb{P}^*$$ and, conversely, every line of$$\mathbb{P}^*$$ corresponds to a point of $$\mathbb{P}^2 (\mathbb{C})$$. Given a curve $$C \subset \mathbb{P}^2 (\mathbb {C})$$, we consider the totality of the tangents to $$C$$: it is a new curve in $$\mathbb{P}^*$$, the so-called dual curve $$C^*$$.

Biduality Theorem: For any projective curve $$C \subset \mathbb{P}^2 (\mathbb {C})$$ we have $$(C^*)^* = C$$. Moreover, if $$p$$ is a simple point of $$C$$ and $$h$$ is a simple point of $$C^*$$, then $$h$$ is tangent to $$C$$ in $$p$$ if and only if $$p$$, considered as a straight line in $$\mathbb{P}^*$$ is tangent to $$C^*$$ in $$h$$.

Now, from the book Discriminants, Resultants, and Multidimensional Determinants (http://books.google.com/books?id=2z...resnum=1&ved=0CBUQ6AEwAA#v=onepage&q&f=false"), reported verbatim:

To give an intuitive sense of Biduality Theorem in $$C \subset {\mathbb{P}^2} (\mathbb {C})$$, we express the notion of tangency in the dual projective plane $$\mathbb{P}^*$$ in terms of the original plane $$\mathbb{P}^2(\mathbb{C})$$. By definition, a tangent to a curve at some point is the line that contains this point and that is infinitely close to the curve near this point.
In our situation, a point of $$\mathbb {P}^*$$ is a line $$l \subset \mathbb{P}^2$$. A curve $$C$$ in $$\mathbb{P}^*$$ is a 1-parameter family of lines in $$C \subset \mathbb {P}^2 (\mathbb{C})$$. A line in $$\mathbb{P}^*$$ is a pencil $$P^*$$ of all the lines in $$C \subset \mathbb{P}^2 (\mathbb{C})$$ for a given point $$P$$ to $$C \subset \mathbb{P}^2 (\mathbb{C})$$. The condition that $$P^*$$ is tangent to $$C$$ in $$l$$ means that the line $$l$$ is a member of the family $$C$$, a point $$P$$ lies on $$l$$ and other lines of $$C$$ in the vicinity of $$l$$ are infinitely close to the pencil $$P^*$$. This is usually expressed by saying that $$P$$ is a caustic point for the family of lines $$C$$.
One can imagine that a beam of light of a certain intensity is coming along each line of $$C$$. Then the total brightness of the incoming light in an arbitrary small neighborhood of a caustic point $$P$$ is infinite, although there is only a ray (line of $$C$$) that meets the point [ tex] P [/tex] itself. The set of all caustic points of the family of lines is usually called the caustic curve [/ b] of $$C$$. This is nothing but the dual projective curve $$C^* \subset \mathbb{P}^2$$.
Then the Biduality theorem states that any curve is the caustic of the family of its tangent lines (envelope of tangents). This is intuitively obvious.
The "dual" form of this theorem is less obvious: it means that every 1-parameter family in $$C$$ of lines in $$\mathbb{P}^2 (\mathbb{C})$$ is the tangent line to any curve in $$\mathbb{P}^2 (\mathbb{C})$$ and this curve is the caustic of $$C$$. An example of a 1-parameter family of straight lines, which is not derived a priori as tangent lines to some curve is given by the reflection of a beam of parallel light in a curved mirror.

So now I ask: the caustic curve we are talking about is the same as the curve that best known in physics is nothing but the envelope of the rays reflected from a curved surface, and coming from a light source? Or maybe the dual of a curve is nothing but the caustic of the curve, caustic in the 'physical' sense given above?
And then, why should it be 'intuitively obvious' that each curve coincides with the caustic of its tangent lines (because what is gathered, the text defines the caustic curve from a family of 1-parameter lines , but the fact that these result in a curve as their envelope is a different kettle of fish)?
Finally, the dual form of the above consideration seems to me sincerely as obvious: trivially, every 1-parameter family of straight lines are tangents to any curve. What's wrong?

 

[mdoni]

Sorry, how could I give the correct Tex-view to my post? I'm afraid, sorry.

 

[mjpam]

Definition: Let $\mathbb{P}^*$ the projective plane dual of $\mathbb{P}^2 (\mathbb{C})$: every line of $\mathbb{P}^2 (\mathbb{C})$ identifies a point of $\mathbb{P}^*$ and, conversely, every line of$\mathbb{P}^*$ corresponds to a point of $\mathbb{P}^2 (\mathbb{C})$. Given a curve $C \subset \mathbb{P}^2 (\mathbb {C})$, we consider the totality of the tangents to $C$: it is a new curve in $\mathbb{P}^*$, the so-called dual curve $C^*$.

Biduality Theorem: For any projective curve $C \subset \mathbb{P}^2 (\mathbb {C})$ we have $(C^*)^* = C$. Moreover, if $p$ is a simple point of $C$ and $h$ is a simple point of $C^*$, then $h$ is tangent to $C$ in $p$ if and only if $p$, considered as a straight line in $\mathbb{P}^*$ is tangent to $C^*$ in $h$.

Now, from the book Discriminants, Resultants, and Multidimensional Determinants (http://books.google.com/books?id=2z...resnum=1&ved=0CBUQ6AEwAA#v=onepage&q&f=false"), reported verbatim:

To give an intuitive sense of Biduality Theorem in $C \subset {\mathbb{P}^2} (\mathbb {C})$, we express the notion of tangency in the dual projective plane $\mathbb{P}^*$ in terms of the original plane $\mathbb{P}^2(\mathbb{C})$. By definition, a tangent to a curve at some point is the line that contains this point and that is infinitely close to the curve near this point.
In our situation, a point of $\mathbb {P}^*$ is a line $l \subset \mathbb{P}^2$. A curve $C$ in $\mathbb{P}^*$ is a 1-parameter family of lines in $C \subset \mathbb {P}^2 (\mathbb{C})$. A line in $\mathbb{P}^*$ is a pencil $P^*$ of all the lines in $C \subset \mathbb{P}^2 (\mathbb{C})$ for a given point $P$ to $C \subset \mathbb{P}^2 (\mathbb{C})$. The condition that $P^*$ is tangent to $C$ in $l$ means that the line $l$ is a member of the family $C$, a point $P$ lies on $l$ and other lines of $C$ in the vicinity of $l$ are infinitely close to the pencil $P^*$. This is usually expressed by saying that $P$ is a caustic point for the family of lines $C$.
One can imagine that a beam of light of a certain intensity is coming along each line of $C$. Then the total brightness of the incoming light in an arbitrary small neighborhood of a caustic point $P$ is infinite, although there is only a ray (line of $C$) that meets the point $P$ itself. The set of all caustic points of the family of lines is usually called the caustic curve of $C$. This is nothing but the dual projective curve $C^* \subset \mathbb{P}^2$.
Then the Biduality theorem states that any curve is the caustic of the family of its tangent lines (envelope of tangents). This is intuitively obvious.
The "dual" form of this theorem is less obvious: it means that every 1-parameter family in $C$ of lines in $\mathbb{P}^2 (\mathbb{C})$ is the tangent line to any curve in $\mathbb{P}^2 (\mathbb{C})$ and this curve is the caustic of $C$. An example of a 1-parameter family of straight lines, which is not derived a priori as tangent lines to some curve is given by the reflection of a beam of parallel light in a curved mirror.

So now I ask: the caustic curve we are talking about is the same as the curve that best known in physics is nothing but the envelope of the rays reflected from a curved surface, and coming from a light source? Or maybe the dual of a curve is nothing but the caustic of the curve, caustic in the 'physical' sense given above?
And then, why should it be 'intuitively obvious' that each curve coincides with the caustic of its tangent lines (because what is gathered, the text defines the caustic curve from a family of 1-parameter lines , but the fact that these result in a curve as their envelope is a different kettle of fish)?
Finally, the dual form of the above consideration seems to me sincerely as obvious: trivially, every 1-parameter family of straight lines are tangents to any curve. What's wrong?

The [noparse][itex][/itex][/noparse] tags give in-line LATEX.

 

[mdoni]

Thanks.
No reply? :(

 

[blue_raver22]

I find this discussion on caustic curves and dual curves to be very interesting. It seems like you are trying to understand and interpret the Biduality Theorem in terms of caustic curves, which are commonly known in physics as the envelope of reflected rays from a curved surface. You are also questioning the intuition behind the theorem and its dual form.

First of all, let me clarify that the caustic curve we are discussing here is different from the physical caustic curve. The caustic curve in projective geometry is defined as the set of all caustic points of a 1-parameter family of lines, while the physical caustic curve is the envelope of reflected rays. However, the concept of caustic points and curves are related in the sense that the caustic curve in projective geometry is the dual of a curve, which is also known as the envelope of tangents.

Now, to address your question about why it is intuitively obvious that each curve coincides with the caustic of its tangent lines, we need to understand the concept of tangency in projective geometry. In this context, a tangent to a curve at a point is defined as the line that is infinitely close to the curve at that point. This means that the tangent line and the curve share the same point, but they are not identical. However, as we take more and more tangent lines at different points on the curve, the envelope of these lines will start to converge and eventually coincide with the curve itself. This is why it is intuitively obvious that a curve is the caustic of its tangent lines.

As for the dual form of the theorem, it is based on the duality principle in projective geometry, which states that there is a correspondence between points and lines in the projective plane. This means that for every point P on a curve C, there is a line in the dual plane that corresponds to it, and vice versa. Therefore, every 1-parameter family of lines in C can be seen as the tangent line to a point on the dual curve C^*. And since the dual curve is defined as the set of all caustic points of a 1-parameter family of lines, it follows that every 1-parameter family of lines in C is also the tangent line to the caustic of C. This is why the dual form of the theorem seems obvious.

In summary, caust