Radius of curvature in polar coordinates is one of the topics in the second module of VTU Engineering Mathematics 1 (BMATS101 under the 2022 scheme, 18MAT11 under 2018). Students find it hard to recall for three reasons: the formula looks similar to the Cartesian version, the algebra is longer, and the same curve can be described in more than one way. This guide derives the working formula, applies it to standard curves, and points out the errors that cost marks.
What Radius of Curvature Measures
At any point on a curve, the radius of curvature ρ is the radius of the circle that best fits the curve at that point. A straight line has infinite radius of curvature because it does not bend. A tight bend, such as the cusp of a cardioid, has a small radius. Engineers use the idea in road and rail design, beam deflection, and any problem where the bending of a path matters.
In Cartesian coordinates the formula is ρ = [1 + (dy/dx)²]^(3/2) / |d²y/dx²|. Polar curves, given as r = f(θ), are more naturally described in polar coordinates, and the formula has to be adapted.
The Polar Formula
Let r be a function of θ. Write r₁ = dr/dθ and r₂ = d²r/dθ². Then the radius of curvature at a point on the curve is:
ρ = (r² + r₁²)^(3/2) / (r² + 2r₁² − r·r₂)
Compute r, r₁ and r₂ first, substitute them into the formula, and simplify. Keep the algebra organised. Most lost marks come from mixing up r₁ and r₂, or dropping a term in the denominator. Some courses also ask for the angle ψ between the radius vector and the tangent, given by tan ψ = r/r₁. Check your syllabus wording.
Worked Example 1: The Cardioid r = a(1 + cos θ)
This curve is a standard exam example, and it shows the formula at work.
- r = a(1 + cos θ)
- r₁ = −a sin θ
- r₂ = −a cos θ
Numerator. r² + r₁² = a²(1 + cos θ)² + a² sin² θ = a²(1 + 2cos θ + cos² θ + sin² θ) = a²(2 + 2cos θ) = 2a·r. So the numerator is (2ar)^(3/2).
Denominator. r² + 2r₁² − r·r₂ = a²(1+cos θ)² + 2a² sin² θ + a²(1+cos θ)cos θ. Expanding and collecting terms gives a²(3 + 3cos θ) = 3a·r.
Result. ρ = (2ar)^(3/2) / (3ar) = (2√2/3)·√(ar).
Check at θ = 0. There r = 2a and r₁ = 0, r₂ = −a. The direct formula gives ρ = (4a²)^(3/2) / (4a² + 2a²) = 8a³ / 6a² = 4a/3. The simplified form (2√2/3)√(a·2a) = (2√2/3)(√2·a) = 4a/3 agrees, so the simplification is correct.
Worked Example 2: The Equiangular Spiral r = aeᶿ
This curve has a special property: the angle between the radius vector and the tangent is constant. Use it as a check on your method.
- r = a eᶿ
- r₁ = a eᶿ = r
- r₂ = a eᶿ = r
Numerator. r² + r₁² = 2r², so the numerator is (2r²)^(3/2) = 2√2 · r³.
Denominator. r² + 2r₁² − r·r₂ = r² + 2r² − r² = 2r².
Result. ρ = 2√2 r³ / 2r² = √2 · r.
For this spiral the radius of curvature is always √2 times the radius vector. That is an easy result to remember as a check.
Worked Example 3: The Circle r = a sin θ
This example is short, and it confirms the formula on a shape whose answer you already know.
- r = a sin θ
- r₁ = a cos θ
- r₂ = −a sin θ
Numerator. r² + r₁² = a² sin² θ + a² cos² θ = a², so the numerator is (a²)^(3/2) = a³.
Denominator. r² + 2r₁² − r·r₂ = a² sin² θ + 2a² cos² θ + a² sin² θ = 2a².
Result. ρ = a³ / 2a² = a/2.
The curve r = a sin θ is a circle of diameter a, so its radius of curvature at every point is a/2. The formula confirms this, which is the reason to check your method on a known shape before trusting it on a new one.
How to Set Up a Polar Curvature Question
- Write r, r₁ and r₂ explicitly. Differentiate r with respect to θ twice, and keep the signs.
- Compute r² + r₁² and simplify. Look for a common factor such as 2ar; spotting it saves work.
- Compute the denominator r² + 2r₁² − r·r₂. Expand carefully, because this is where most errors appear.
- Substitute and simplify. Keep the powers in the numerator and denominator organised.
- Check at a convenient point. Choose θ = 0 or θ = π, where the calculation is simplest, and verify the value.
Common Mistakes That Cost Marks
Mistake 1: Mixing up r₁ and r₂ in the denominator. The formula uses both with different coefficients. Mixing them gives an answer that looks reasonable and is wrong.
Mistake 2: Dropping the sign of r₂. For the cardioid, r₂ = −a cos θ. Losing the minus sign changes the denominator and the result.
Mistake 3: Getting the 3/2 power wrong. The numerator has the power 3/2 applied to the whole sum, not a cube.
Mistake 4: Forgetting that ρ is a length. The radius of curvature should be positive. A negative result means a sign error in r₂ or in the denominator.
Practice Questions
- Find the radius of curvature of the circle r = a sin θ at any point. (Expected: a/2, as in Worked Example 3.)
- Find the radius of curvature of the cardioid r = a(1 − cos θ) at θ = π. (Hint: r = 2a, r₁ = 0, r₂ = −a. Expected: 4a/3.)
- Find the radius of curvature of the Archimedean spiral r = aθ at a general point. (Hint: r₁ = a and r₂ = 0. The result simplifies to ρ = a(1 + θ²)^(3/2) / (θ² + 2). At θ = 0 this gives a/2.)
Try each question fully before reading the hints, and compare your final expression with the expected result.
Using Known Shapes to Build Confidence
When a polar formula is new, it helps to test it on a curve whose answer you can predict. The circle r = a cos θ is a good test. Here r = a cos θ, r₁ = −a sin θ, and r₂ = −a cos θ.
Numerator. r² + r₁² = a² cos² θ + a² sin² θ = a², so the numerator is a³.
Denominator. r² + 2r₁² − r·r₂ = a² cos² θ + 2a² sin² θ + a² cos² θ = 2a².
Result. ρ = a³ / 2a² = a/2.
The circle r = a cos θ has diameter a, so its radius of curvature at every point must be a/2. The formula confirms it. Notice that this is the same answer as for r = a sin θ, because both are circles of the same size, only rotated. If the formula gave a different answer for a circle, the error would be in the algebra, not the method.
Where Students Lose Marks in Long Questions
In a long polar question, the risk is accumulated error. A small slip in r₂ at the start carries through the denominator and changes the final expression. Three habits reduce this risk. First, write r, r₁, and r₂ on separate lines and check each against the original function before substituting. Second, simplify the numerator and denominator separately, and write each intermediate result. Third, evaluate the final expression at one convenient angle, such as θ = 0 or θ = π/2, and compare it with a direct calculation. A mismatch shows exactly where to look.
Connecting This Topic to the Full Module
Radius of curvature in polar coordinates is taught alongside polar curves and partial differentiation in the second module of VTU Engineering Mathematics 1. For the full module outline, the topics that are most often tested, and the places where marks are usually lost, see our VTU Engineering Maths 1 Notes — Complete Module-Wise Guide (BMATS101).
Frequently Asked Questions
Q: Is there a shorter formula for polar curves?
The general formula above is the standard one. Shortcuts exist for particular curves, but learning the general formula is more reliable in an exam.
Q: Why does the formula contain r₂?
The curvature depends on how the direction of the curve changes, and r₂ carries that information in polar form.
Q: Where does this fit in the module?
It is taught with polar curves and partial differentiation in the second module. The full syllabus map is in our VTU Engineering Maths 1 notes guide.
For fully solved polar curve problems and formula sheets, see the Engineering Math Survival Kit. To ask about batches covering this module, use the contact page.