Separate signed radius, distance and estimates

Try

For the limacon r=1+2cosθ, move from θ=2π/3 to θ=π. The signed radius falls from 0 to −1, yet the point moves from the origin to (1,0): its distance from the origin grows from 0 to 1. “Radius decreases” is therefore incomplete unless you identify whether you mean signed r or nonnegative distance ∣r∣.

The two quantities share the same angle input but not the same vertical values. In the supplied graph, the signed-radius trace passes below zero between the two origin angles. The distance trace reflects only that negative portion upward. Everywhere r≥0, the traces coincide. This is a graph against angle, not the Cartesian shape of the limacon.

Predict

Predict every nearest and farthest angle on 0≤θ≤2π. The minimum signed radius is −1 at π, but that is not the minimum distance. Distance zero occurs at the origin angles 2π/3 and 4π/3. The global farthest distance is 3 at 0 and 2π, which represent the same Cartesian point. At π the inner loop has its own local distance maximum of 1.

Angle interval	Signed r trend	Sign of r	Distance ∣r∣ trend
(0,2π/3)	Decreasing	Positive	Decreasing
(2π/3,π)	Decreasing	Negative	Increasing
(π,4π/3)	Increasing	Negative	Decreasing
(4π/3,2π)	Increasing	Positive	Increasing

Cosine decreases from 1 to −1 over [0,π] and increases back to 1 over [π,2π]. Combine that monotonicity with ∣r∣=r when r≥0 and ∣r∣=−r when r<0. This proves the complete interval table without derivatives. A sampled animation is useful for checking your interpretation but cannot replace the sign argument.

Experiment

Now use a table without a supplied generating function. The two observations are (θ₀,r₀)=(0,2) and (θ₁,r₁)=(π,−2). Both are the same Cartesian point (2,0), and both distances are 2. Predict what linear signed-radius interpolation gives at θ=π/2 before pressing the button.

Interpolate signed radius carefully

Separate workflow: interpolate signed radius

Enter two angular/radial observations and an interior query. This constructs a straight-line estimate of signed r versus θ; it is not distance interpolation, a Cartesian chord interpolation, or a guaranteed value of a polar curve model. These observations define their own interpolation exercise independently of any curve exploration.

First angle θ₀/π (−2 to 2)First signed radius r₀ (−10 to 10)Second angle θ₁/π (−2 to 2)Second signed radius r₁ (−10 to 10)Query angle θ/π (strictly between endpoints)

Interpolate signed radius

Reset polar workbench

All controls are manual; nothing animates automatically. Editing a trace input clears the trace; editing an interpolation input clears that estimate. Reset restores every input and hides calculations. This temporary workbench does not save work, mark completion or award mastery. Retain your predictions, coordinate evidence and assumptions in the lesson response.

Keep First angle at 0, First signed radius at 2, Second angle at 1, Second signed radius at −2 and Query angle at 0.5. The angle fields use multiples of π, so the second angle 1 means π radians. Select Interpolate signed radius. The halfway signed estimate is 0, so its estimated distance is 0 and its Cartesian point is the origin.

Three different assumptions give three different conclusions. Interpolating signed r then taking its absolute value gives distance 0. Interpolating the endpoint distances directly gives distance 2; if you then place that distance on the query ray, the point would be (0,2). Interpolating Cartesian endpoint coordinates gives (2,0), since those endpoints coincide. These are different models of what happens between observations, not three numerical implementations of the same instruction. The table alone selects none of them.

Understand

For two ordered angles θ₀<θ₁ and a query strictly between them, the fractional position is λ=(θ−θ₀)/(θ₁−θ₀). Linear signed interpolation assumes

rhat=(1−λ)r₀+λr₁.

Then estimate distance by ∣rhat∣ and coordinates by (rhat cosθ,rhat sinθ). Use the actual angular gap. For angles π/6 and 2π/3, their halfway angle is 5π/12, not π/3. The workbench uses finite bounded inputs and refuses queries outside or at its interpolation endpoints; this strict-interior rule is a tool contract, while the algebraic linear formula also evaluates at endpoints.

The absolute-value operation is nonlinear through a sign change. In general ∣(1−λ)r₀+λr₁∣ differs from (1−λ)∣r₀∣+λ∣r₁∣. It agrees when the endpoint radii have the same sign and 0≤λ≤1, but that algebraic fact still does not prove the true unknown radial function follows either interpolating line. A prediction rule and an observed curve are separate evidence.

No finite error certificate follows from two exact samples and continuity alone. For the default endpoints, the straight radial rule is L(θ)=2−4θ/π. Add a bump B(θ)=cθ(π−θ). It vanishes at both endpoints, so R=L+B matches the table for every real c. At π/2, R=cπ²/4. Choosing c=12/π² gives 3 instead of 0, while arbitrary c permits arbitrarily large discrepancies. Each alternative is continuous. Additional model or regularity information is needed to bound interpolation error; the widget deliberately reports an estimate rather than an accuracy guarantee.

Apply

Enter a changed bracket: θ₀=π/4,r₀=−2 and θ₁=3π/4,r₁=4. Query θ=π/2. Record the interpolation inputs and compute the signed estimate, distance and Cartesian point before revealing the tool result. Explain whether a positive change in signed radius implies moving farther from the origin over the whole bracket.

Check the changed estimate

The fraction is 1/2, so rhat=1, distance 1 and estimated point (0,1). The assumed signed radius increases from −2 to 4 but passes through zero, so its distance first decreases and then increases. Its linear zero occurs when −2+6λ=0, hence λ=1/3 and θ=5π/12. This crossing angle belongs to the assumed interpolating rule, not to every continuous radial function matching the endpoints.

A change in radius per radian is not physical speed. For the same bracket, average signed change is 6/(π/2)=12/π coordinate units per radian. Endpoint distances change from 2 to 4, giving average distance change 4/π per radian. Neither is arc length traveled, and no time law has been supplied. Even if θ were later tied to time, Cartesian motion includes direction change as well as radial change.

For the rose r=2cos(2θ), use exact zeros and sign intervals to find all distance extrema on [0,2π]. The nearest points occur at π/4+kπ/2 for k=0,1,2,3. Farthest distance 2 occurs at 0,π/2,π,3π/2,2π, including negative-radius tips. Check one negative interval by hand and describe how decreasing r can coincide with increasing distance.

Continue

Keep the scalar quantity named in every claim: signed radius, distance from the origin, a Cartesian coordinate, displacement, or travel length. Retain the angular interval and units, the interpolation assumption, and any uncertainty information actually supplied. A small table cannot justify an unreported precision bound or identify an entire trajectory.

Return to complete signed-radius curve construction when a curve formula is known. In that case exact zeros, signs and unit-circle monotonicity can support complete interval arguments. When only samples are known, estimates remain conditional and an unused measurement can challenge the chosen interpolation model.