Envelope of Parabola Tangents
Compare neighboring tangent lines and watch their intersection approach the parabola they envelope.
Free theory articleCompare neighboring tangent lines and watch their intersection approach the parabola they envelope.
Concept & interpretation
The envelope of a family of lines is a curve that is tangent to each member of the family at some point. By examining the tangent lines of a parabola, we can observe how the intersection of two neighboring tangent lines approaches the parabola itself as their parameter difference goes to zero.
Interactive Experiment
You can visualize this process using the interactive module at https://riano.app/module/parabola-tangent-envelope.
- Select Parabola coefficient a to 1.
- Set Tangent parameter t to 1.
- Set Neighbor spacing h to 0.5, then decrease it to 0.25 and observe the positive gap decrease from 0.0625 to 0.015625. The module calculates the intersection using the finite h value.
- Observe the Vertical gap value. As h gets smaller, the gap decreases, showing that the intersection approaches the parabola (the envelope).