01 Polar Coordinates and Conic Sections
Polar Coordinates¶
In polar coordinate system, we locate a point with reference to:
- pole a fixed point (usually fixed at the origin)
- initial ray a fixed line, passing through the pole (usually \(+x\) axis)
Let
- \(r\) - directed distance of the point from pole
- \(r > 0\) forward
- \(r < 0\) backward
- \(\theta\) - directed angle of radius vector from the initial ray
- \(\theta < 0\) anti-clockwise
- \(\theta > 0\) clockwise
- \(P(r, \theta)\) - corresponding point
Circle Through Pole¶
\[ r = \pm a, \quad 0 \le \theta \le 2 \pi \]
represents a circle with center @pole and radius \(a\). Sign can be either, because it is the same circle traversed in the opposite direction
Straight line through pole¶
\[ \theta = \theta_0, \quad - \infty < r < \infty \]
IDK¶
| \(r\) | \(\theta\) | Diagram |
|---|---|---|
| const | const | point |
| const | inequality | arc |
| inequality | const | straight line segment |
| inequality | inequality | region |
Cartesian \(\iff\) Polar¶
Consider the point \(P(x, y) \iff P(r, \theta)\)
\[ \begin{aligned} x &= r \cos\theta \\ y &= r \sin\theta \\ r^2 &= x^2 + y^2 \\ \theta &= \tan^{-1} \left( \frac y x \right) \end{aligned} \]
Symmetry¶
Let \(r = f(\theta)\) be a polar curve
X-axis¶
\(P(r, \theta)\) and \(P'(r, - \theta)\) lie on same graph
| Symmetry about | Vary theta | \(P(r, \theta)\) lies on the same graph as | or \(P(r, \theta)\) lies on the same graph as |
|---|---|---|---|
| X-axis | \(0 \le \theta \le \pi\) | \(P'(r, -\theta)\) | \(P'(-r, \pi -\theta)\) |
| Y-axis | \(\frac{-\pi} 2 \le \theta \le \frac \pi 2\) | \(P'(-r, -\theta)\) | \(P'(r, \pi -\theta)\) |
| Origin | \(0 \le \theta \le \frac \pi 2\) | \(P'(-r, \theta)\) | \(P'(r, \pi + \theta)\) |
Shapes¶
Limacon¶
\[ r = a \pm b \cos\theta \\ \text{ or } \\ r = a \pm b \sin\theta \]
| \(\frac a b\) | Type |
|---|---|
| \(<1\) | inner loop |
| \(=1\) | cardioid |
| \(>1\) | outer loop |
Roses¶
\[ \begin{aligned} r &= a \cos(n\theta) \\ &\text{ or } \\ r &= a \sin(n\theta) \\ \text{No of petals } N &= \begin{cases} n, & n = \text{odd} \\ 2n, & n = \text{even} \end{cases} \\ \text{Axis of first petal } \theta &= \begin{cases} 0 & r = a \textcolor{orange}{\cos}(n \theta) \\ \dfrac \pi {2n} & r = a \textcolor{orange}{\sin} (n \theta) \end{cases} \\ \text{Length of petals} &= a \\ \text{Angular Gap between axes of petals} &= \frac{2 \pi}{N} \end{aligned} \]
Lemmiscates¶
\[ r^2 = a \cos\theta \\ \text{ or } \\ r^2 = a \sin\theta \\ \]
Straight Line¶
\[ r \cos(\theta-\theta_0) = r_0 \]
- \(P(r, \theta)\) is any point on given line
- \(P_0(r_0, \theta_0)\) is foot of \(\perp\)r from the pole
Circle¶
\[ r^2 + {r_0}^2 - 2 r r_0 \cos(\theta - \theta_0) = a^2 \\ \]
- \(P(r, \theta)\) is any point on circle
- \(P_0(r_0, \theta_0)\) is center of circle
- \(a\) is radius
Radius passing through pole¶
\[ r_0 = a\\ r = 2a cos(\theta - \theta_0) \]
Center lies on axis¶
| Center at | \(r\) |
|---|---|
| \((a,0)\) | \(2a \cos \theta\) |
| \((-a,0)\) | \(-2a \cos \theta\) |
| \((a, \frac \pi 2)\) | \(2a \sin \theta\) |
| \((a, -\frac \pi 2)\) | \(-2a \sin \theta\) |
Area under curve¶
For a polar curve \(r = f(\theta), \alpha \le \theta \le \beta\)
\[ A = \frac12 \int\limits_{\theta = \alpha}^\beta r^2 \cdot d\theta \]
For area bounded by the curves \(r_1 = f_1(\theta), r_2 = f_2(\theta), \alpha \le \theta \le \beta\) such that \(r_1 < r_2\)
\[ A = \frac12 \int\limits_{\theta = \alpha}^\beta {r_2}^2 - {r_1}^2 \cdot d\theta \]
Length of curve¶
For a curve \(r = f(\theta), \alpha \le \theta \le \beta\) traversed exactly once from \(\theta = \alpha \to \beta\)
\[ L = \int\limits_{\theta = \alpha}^\beta \sqrt{ r^2 + (r')^2 } \cdot d\theta \qquad \left[ r' = \frac{dr}{d \theta} \right] \]
Conic Sections¶
Let
- \(P(r, \theta)\) be any point on the conic section with focus at origin
- \(e = \dfrac{ \text{Distance bw focii} }{ \text{Distance bw vertices} }\)
| Directrix | \(r\) |
|---|---|
| \(x = a\) | \(\frac{ke}{1 + e \cos\theta}\) |
| \(x = -a\) | \(\frac{ke}{1 - e \cos\theta}\) |
| \(y = a\) | \(\frac{ke}{1 + e \sin\theta}\) |
| \(y = -a\) | \(\frac{ke}{1 - e \sin\theta}\) |
Shapes¶
| \(e\) | Shape |
|---|---|
| \(0 < e < 1\) | Ellipse |
| \(e = 1\) | Parabola |
| \(e > 1\) | Hyperbola |
For ellipse,
\[ k = a \left[ \frac 1 e - e \right] \]