class: middle, center # Polarity and Conic Wai-Shing Luk 2017-06-24 📅 --- ### Polarities - A *polarity* is a projective correlation of period 2. - We call $a$ the *polar* of $A$, and $A$ the pole of $a$. - Denote $a = A^\perp$ and $A = a^\perp$. - Except degenerate cases, $A = (A^\perp)^\perp$ and $a = (a^\perp)^\perp$. - It may happen that $A$ is incident with $a$ so that each is *self-conjugate*. - The locus of self-conjugate points defines a *conic*. However, the polarity is a more general concept than a conic, because some polarities may not have self-conjugate points (or their self-conjugate points are complex). --- ### The Use of a Self-Polar Triangle - Any projective correlation that relates the three vertices of one triangle to the respectively opposite sides is a polarity. - Thus, any triangle $\\{ABC\\}$, any point $P$ not on a side, and any line $p$ not throughout a vertex, determine a definite polarity $(ABC)(Pp)$. --- ### The Conic - Historically *ellipse* (including *circle*), *parabola*, and *hyperbola*. - The locus of self-conjugate points is a *conic*. - Their polars are its *tangents*. - Any other line is called a *secant* or a *nonsecant* according to as it meets the conic twice or not at all, i.e., according to as the involution of conjugate points on it is hyperbolic or elliptic. - Note: Intersecting a conic with a line may result of an irrational intersection point. --- ### 🔄 Conic & Polarity .mermaid[ <pre> graph LR P[Point] -->|polar| L[Polar Line] L -->|pole| P C[Conic] -->|self-conjugate| S["Self-Conjugate\n Points"] S -->|tangent| T[Tangent Line] style P fill:#4caf50 style L fill:#2196f3 style C fill:#ff9800 style S fill:#9c27b0 style T fill:#f44336 </pre> ] --- ### Construct the polar of a point using a conic - To construct the polar of a given point $C$, not on the conic, draw any two secants $PQ$ and $RS$ through $C$; then the polar joins the two points meet($QR, PS$) and meet($RP, QS$). --- ### 📚 Example of constructing the polar of a point \begin{figure}[hp] \centering \input{pole2polar.tikz} \caption{Example of constructing the polar of a point} \end{figure} --- ### Another example of constructing the polar of a point \begin{figure}[hp] \centering \input{pole2polar2.tikz} \caption{Another example of constructing the polar of a point} \end{figure} --- ### Construct the pole from a line - To construct the pole of a given secant $a$, draw the polars of any two points on the line; then the common point of two polars is the pole of $a$. --- ### Constructing the pole of a line \begin{figure}[hp] \centering \input{polar2pole.tikz} \caption{Constructing the pole of a line} \end{figure} --- ### Construct the tangent of a point on a conic - To construct the tangent at a given point $P$ on a conic, join $P$ to the pole of any secant through $P$. --- ### 📚 Example of construct the tangent of a point on a conic \begin{figure}[hp] \centering \input{tangent.tikz} \caption{Construct the tangent of a point on a conic} \end{figure} --- ### Another example of constructing the tangent of a point on a conic \begin{figure}[hp] \centering \input{tangent2.tikz} \caption{Another example of constructing the tangent of a point on a conic} \end{figure} --- ### Pascal's Theorem - If a hexagon is inscribed in a conic, the three pairs of opposite sides meet in collinear points. --- ### An instance of Pascal' theorem {#fig:pascal} --- ### Another instance of Pascal' theorem {#fig:pascal2} --- ### Backup [melpon.org](http://melpon.org/wandbox/permlink/xUFygXm8ytQNwUJ0) ``` {.terminal} > pandoc -s --mathjax -t revealjs -V theme=default -o proj_geom.html proj_geom.md > pandoc -t beamer -o proj_geom.svg proj_geom.md beamer.yaml > pandoc -o proj_geom.docx proj_geom.md ```