Find The Polar Equation Of The Conic Calculator

Find the Polar Equation

Enter the eccentricity and directrix details to find the polar equation of the conic section (parabola, ellipse, or hyperbola) with a focus at the origin.

What is a Polar Equation of a Conic Calculator?

A polar equation of a conic calculator is a tool used to determine the equation of a conic section (ellipse, parabola, or hyperbola) in the polar coordinate system, given its eccentricity and the location of its directrix relative to a focus at the origin. The standard form of such an equation is r = ep / (1 ± e cos θ) or r = ep / (1 ± e sin θ), where ‘e’ is the eccentricity and ‘|p|’ is the distance from the focus to the directrix.

This calculator is useful for students studying analytic geometry, physics (e.g., orbital mechanics), and engineering, where conic sections and their polar representations are frequently encountered. It simplifies the process of deriving the specific equation based on the conic’s geometric properties.

Common misconceptions include thinking that all conic sections are centered at the origin in this form (only one focus is at the origin) or that ‘p’ is always positive in the formula (it represents a distance, but its sign is incorporated into the directrix form x=±p or y=±p).

Polar Equation of a Conic Formula and Mathematical Explanation

A conic section can be defined as the locus of points P such that the ratio of the distance from P to a fixed point F (the focus) to the distance from P to a fixed line L (the directrix) is a constant ‘e’ (the eccentricity).

If the focus F is at the origin (0,0) of the polar coordinate system, and the directrix L is a line perpendicular or parallel to the polar axis, the polar equation of the conic can be derived.

Let the distance from the focus (origin) to the directrix be |p|. The equation of the directrix can be x=p, x=-p, y=p, or y=-p.

1. Directrix x = p (right of focus): The distance from a point (r, θ) to the directrix is p – r cos θ. The definition of a conic is r / |p – r cos θ| = e. Assuming p > 0 and 1 + e cos θ > 0, r = e(p – r cos θ) => r(1 + e cos θ) = ep => r = ep / (1 + e cos θ).
2. Directrix x = -p (left of focus): The distance from (r, θ) to x=-p is p + r cos θ (for p>0). r = e(p + r cos θ) => r = ep / (1 – e cos θ) (using |p|, directrix at -|p| implies |p|+rcosθ, so r=e(|p|+rcosθ) if we take p as |p|, this leads to 1-ecos with x=-p being |p| to the left, so dist is |p|+rcos, but the formula uses 1-e cos theta for x=-p where p is distance |p|. If x=-d, d>0, dist is d+rcos. r/ (d+rcos) = e, r=ed+ercos, r(1-ecos)=ed, r=ed/(1-ecos). So if directrix is x=-p, p>0, r=ep/(1-ecos))
3. Directrix y = p (above focus): The distance from (r, θ) to y=p is p – r sin θ. r = e(p – r sin θ) => r = ep / (1 + e sin θ).
4. Directrix y = -p (below focus): The distance from (r, θ) to y=-p is p + r sin θ. r = e(p + r sin θ) => r = ep / (1 – e sin θ).

Where:

• r is the distance from the focus (origin) to a point on the conic.
• θ is the angle between the polar axis and the line segment from the origin to the point.
• e is the eccentricity (e ≥ 0).
• |p| is the absolute distance from the focus to the directrix (p > 0).

The type of conic is determined by ‘e’:

• If e = 0, it’s a circle (but our formula assumes p is finite, so e=0 implies r=0 unless p is infinite, so typically we start with e>0 for non-degenerate conics with a directrix). If e is very close to 0, it’s an ellipse close to a circle.
• If 0 < e < 1, it's an ellipse.
• If e = 1, it’s a parabola.
• If e > 1, it’s a hyperbola.

Variables Table

Variable	Meaning	Unit	Typical Range
e	Eccentricity	Dimensionless	e ≥ 0
|p|	Distance from focus to directrix	Length units	|p| > 0
r	Distance from focus to point on conic	Length units	r ≥ 0 or undefined
θ	Angle in polar coordinates	Radians or Degrees	0 ≤ θ < 2π (or 0° ≤ θ < 360°)

Variables used in the polar equation of a conic.

Practical Examples

Example 1: Parabola

Suppose we have a conic with eccentricity e = 1 and a directrix x = -3 (left of the focus at the origin, so |p|=3).

• e = 1
• |p| = 3
• Directrix: x = -3 (corresponds to x = -p, so 1 – e cos θ denominator)
• ep = 1 * 3 = 3
• Equation: r = 3 / (1 – 1 cos θ) = 3 / (1 – cos θ)

This is the polar equation of a parabola opening to the right.

Example 2: Ellipse

Consider an ellipse with eccentricity e = 0.5 and a directrix y = 4 (above the focus at the origin, so |p|=4).

• e = 0.5
• |p| = 4
• Directrix: y = 4 (corresponds to y = p, so 1 + e sin θ denominator)
• ep = 0.5 * 4 = 2
• Equation: r = 2 / (1 + 0.5 sin θ)

This is the polar equation of an ellipse.

How to Use This Polar Equation of a Conic Calculator

1. Enter Eccentricity (e): Input the non-negative value for the eccentricity ‘e’. Remember e=1 for a parabola, 0 < e < 1 for an ellipse, e > 1 for a hyperbola, and e=0 for a circle (though the standard form with directrix usually applies to e>0).
2. Enter Distance |p|: Input the positive distance |p| from the focus (origin) to the directrix.
3. Select Directrix Type: Choose the form of the directrix equation from the dropdown menu (x=p, x=-p, y=p, or y=-p). This determines whether cos θ or sin θ is used, and the sign in the denominator.
4. Calculate: The calculator automatically updates the results as you input values. You can also click “Calculate”.
5. View Results: The primary result is the polar equation. You’ll also see the conic type, the value of ‘ep’, and the denominator form. A table of ‘r’ values for various angles ‘θ’ is also shown.
6. Reset: Click “Reset” to return to default values.
7. Copy Results: Click “Copy Results” to copy the equation and key values to your clipboard.

The polar equation of a conic calculator helps visualize how ‘e’ and ‘p’ define the shape and size of the conic.

Key Factors That Affect Polar Equation Results

1. Eccentricity (e): Directly determines the type of conic (circle, ellipse, parabola, hyperbola) and its “flatness” or “openness”. Higher ‘e’ means a more open or less circular conic.
2. Distance |p|: Scales the size of the conic. A larger |p| for a given ‘e’ results in a larger conic. ‘ep’ is the numerator, scaling ‘r’.
3. Directrix Location (x=p, x=-p, y=p, y=-p): Determines the orientation of the conic relative to the focus at the origin and whether the equation involves cos θ or sin θ, and the sign in the denominator.
   • x=p or x=-p: Axis of symmetry is along the polar axis (x-axis).
   • y=p or y=-p: Axis of symmetry is perpendicular to the polar axis (y-axis).
4. Sign in the Denominator (1 ± e cos θ or 1 ± e sin θ): Depends on which side of the focus the directrix lies (x=p vs x=-p, or y=p vs y=-p). This affects the direction the conic “opens” or is elongated towards.
5. Focus at Origin: This calculator assumes one focus is at the origin (0,0) of the polar coordinate system, which is standard for this form of the equation.
6. Angle θ: The value of r changes with θ, tracing out the conic. For some angles, r might be undefined (e.g., when the denominator is zero in hyperbolas).

Understanding these factors is crucial when using the polar equation of a conic calculator and interpreting its results.

Frequently Asked Questions (FAQ)

What is eccentricity ‘e’?

Eccentricity is a non-negative number that defines the shape of a conic section. It’s the ratio of the distance from any point on the conic to the focus, to the distance from that point to the directrix.

What is the directrix?

The directrix is a fixed line used in the definition of a conic section. The conic is the set of points where the ratio of distances to the focus and directrix is constant (the eccentricity).

What if e = 0?

If e=0, the equation becomes r=0, which is just the origin, assuming p is finite. A circle with radius ‘a’ centered at the origin has polar equation r=a, which doesn’t directly fit the r=ep/(1±ecosθ) form unless p is infinite or e is near zero with ep=a.

Can ‘p’ be negative?

In our calculator, we ask for |p|, the distance, which is positive. The sign is handled by selecting the directrix type (e.g., x=-p implies the directrix is at x=-|p|).

Why is the focus at the origin?

The standard polar form r = ep/(1 ± e cos θ) or r = ep/(1 ± e sin θ) is derived assuming one focus is at the origin of the polar coordinate system. Conics can be placed elsewhere, but their equations become more complex.

How does the directrix type affect the equation?

If the directrix is x=±p (vertical line), the equation involves cos θ. If it’s y=±p (horizontal line), it involves sin θ. The sign in the denominator depends on whether it’s x=p or x=-p (or y=p vs y=-p).

Can ‘r’ be negative or undefined?

In standard polar coordinates, r is usually non-negative. However, when 1 ± e cos θ or 1 ± e sin θ becomes zero or negative, ‘r’ from the formula can be negative or undefined. For hyperbolas, the denominator can become zero, leading to undefined r (asymptotes), and ‘r’ can be negative, tracing the other branch if we allow r<0.

What are the units of ‘r’ and ‘p’?

They are units of length, and they should be consistent (e.g., both in meters, centimeters, or inches). Eccentricity ‘e’ is dimensionless.