Polar coordinates
Introduced| English | Meaning |
|---|---|
| Cartesian/kɑːˈtiːzɪən/ | Cartesian |
| polar coordinates/ˈpəʊlə kəʊˈɔːdɪnəts/ | Coordinates specifying a point by its distance from an origin and its angle from a reference direction. |
| rotational symmetry/rəʊˈteɪʃənl ˈsɪmətri/ | The property of a shape that matches itself after a rotation of less than one full turn. |
| cardioid/ˈkɑːdɪəʊɪd/ | cardioid |
| sector/ˈsektə/ | A region of a circle bounded by two radii and the arc between them. |
The coordinate system that simplifies circles
- Describing a spiral or a circle in Cartesian 笛卡尔 coordinates requires messy equations. In polar coordinates 极坐标, a circle is just $r = \text{constant}$.
- Polar coordinates are natural for anything with rotational symmetry 旋转对称 — from planetary orbits to radar screens.
Polar coordinates
- Polar coordinates give a point by distance $r$ from the origin and angle $\theta$.
- Links to Cartesian:
Worked example. The point $(r, \theta) = (5, 60°)$ in Cartesian: $x = 5\cos 60° = 2.5$, $y = 5\sin 60° = 4.33$. So the point is $(2.5, 4.33)$.
Polar coordinates $(r,\theta)$ convert to Cartesian by $x=r\cos\theta$ and $y=r\sin\theta$
A spiral staircase: spirals are described naturally in polar coordinates (r, theta)
Polar coordinates
r = f(θ)
In polar form a point is (r, θ). Beautiful curves — roses, cardioids, spirals — come from r = f(θ).
In polar coordinates, the x-coordinate is given by:
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x = r cos θ and y = r sin θ.
Convert (r, θ) = (5, 60°) to Cartesian. What is x = r cos 60°?
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x = 5 cos 60° = 5 × 0.5 = 2.5.
For the point (3, 4) in Cartesian, r² = x² + y². Find r².
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r² = 3² + 4² = 9 + 16 = 25.
Polar curves
- Common polar curves:
r = 4cos θ is a circle of radius 2 centred at (2, 0) — polar form makes it a one-line equation.
- $r = a$ — a circle of radius $a$ centred at the origin.
- $r = a\cos\theta$ — a circle of radius $\dfrac{a}{2}$ centred at $\left(\dfrac{a}{2}, 0\right)$.
- $r = a(1 + \cos\theta)$ — a cardioid 心脏线 (heart shape).
The polar curve r = 4cos θ is a circle. What is its radius?
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r = 4cos θ gives x² + y² = 4x, i.e. (x−2)² + y² = 4 — a circle of radius 2.
The polar curve r = a(1 + cos θ) is called a:
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r = a(1 + cos θ) traces a heart-shaped curve called a cardioid.
Polar area
- Sector 扇形 area: $\text{area} = \tfrac12\displaystyle\int_{\alpha}^{\beta} r^2\,d\theta$.
Don't forget the $\dfrac{1}{2}$. The polar area formula has a factor of $\dfrac{1}{2}$ that's easy to miss. It comes from the area of a thin triangular sector: $\dfrac{1}{2}r^2\,d\theta$.
A cardioid r = 1 + cos theta, a curve sketched directly in polar form
The area of a polar sector is ½∫r² dθ.
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This is the standard polar area formula.
Converting curves
- Example: $r = 4\cos\theta$ → multiply by $r$: $r^2 = 4r\cos\theta$ → $x^2 + y^2 = 4x$ → $(x-2)^2 + y^2 = 4$, a circle radius $2$.
You've got it
- $x = r\cos\theta$, $y = r\sin\theta$, $r^2 = x^2 + y^2$
- polar sector area $= \tfrac12\int r^2\,d\theta$
- $r = 4\cos\theta$ is a circle of radius 2 centred at $(2, 0)$