Dr. Loveless Curiosity Lab

Minimum Distance to a Curve

Student investigator: Tessa

Suppose we have a fixed point and a curve. Which point on the curve is closest to the fixed point?

This is an important type of optimizatio problem. Turn a geometric question into a distance function, then find where that function is smallest.

The first example uses a parametric heart curve in the plane. The second uses the 3D Viviani curve, which is the curve of intersection of a cylinder and a sphere. In both cases, we scan along the curve and watch the distance graph reveal the minimum.

Back to Project Showcase

Example 1: 2D Heart Curve

Move the point \((A,B)\), by clicking on it in the graph and dragging it. Then click "Scan for Minimum Distance". Just for fun, click on the "Make It a Beating Heart". Also watch how the corresponding distance function changes below.

Example 2: 3D Viviani Curve

Adjust the point \((A,B,C)\), by moving the sliders below. Then click "Scan for Minimum Distance". Watch how the distance graph changes.

\(A\) -0.67
\(B\) 0.45
\(C\) 0.16
2D Heart Curve
Move \((A,B)\). The closest point appears after the scan and stays visible until reset or a new scan.
3D Viviani Curve
Use the sliders to move \((A,B,C)\). The closest point appears after scanning.
Distance vs \(t\) Along the Heart
Distance from \((A,B)\) to \((X(t),Y(t))\).
Distance vs \(t\) for the Viviani Curve
Distance from \((A,B,C)\) to \((X(t),Y(t),Z(t))\).

Where It Started

Tessa started with a Math 124 optimization problem about finding the minimum distance from a curve to a point.

The homework included examples such as finding the minimum distance from a line to the origin and finding the closest point on another curve. This project takes that familiar calculus problem and turns it into a visual exploration.

Instead of only solving one equation, we can watch the distance change as a point moves along the curve.

Going Further

These questions are everything you look... As you walk around campus which point on a nearby building is closest to you? Which point on Earth is closest to a satellite? How does a robot vacuum find nearby obstacles so it does not run into them? How does a boat navigate a narrow stream while staying away from the banks?

The same idea returns in Math 126 when students find the closest point on a surface to a given point.

Further Reading