The answer, as with most interesting problems, is: it depends. And exploring that dependency reveals a beautiful interplay of geometry that has deep implications in fields from physics to game design. With the interactive analysis tool we've built, you can see this duel of curves play out in real-time.

What is "Curvature"?
Our intuitive sense of "curviness" has a precise mathematical definition: curvature. It's not just the slope of a line, but the rate at which the slope itself is changing.
Imagine you're driving a car. The steering wheel's position is the slope. Curvature is how quickly you have to turn the steering wheel to stay on the road. A gentle highway bend has low curvature, while a sharp hairpin turn has extremely high curvature.

A Guided Tour of the Analysis Tool
The best way to understand the relationship between these two shapes is to play with them. Our interactive tool is designed to let you do just that.

1. The Main Event: The Curves
When you first load the tool, you'll see the two main actors:
A cyan-colored sine wave arc.
A magenta-colored parabola arc.
The lines connecting them are colored based on the difference in their curvature at each point. Yellow means a huge difference; purple means they are very similar. Notice the distinctive "U" shape of the color pattern—the biggest difference is right in the middle!

2. The Control Panel & Metrics
On the side, you have the control panel. The sliders let you change the Sine Amplitude (how tall the sine wave is) and the Parabola Height. As you change these, all the metrics and graphs update instantly.
The Global Metrics give you the big picture:
Arc Length: The total length of each curve from start to finish.
Area Between Curves: The total area of the space between the two shapes.
Avg. Curvature Diff: A single number representing the average "disagreement" between the two curves across the entire arc. Try to minimize this value to find the "best fit"!

3. The Visualizations: Seeing the Data
This is where the story unfolds.
Curvature Comparison Graph: This is the heart of our question. It plots the curvature of the sine wave (cyan) and the parabola (magenta) directly. You can immediately see how they trade places.
Difference Graph: This shows the absolute difference between the two lines on the graph above. The peak of this graph corresponds to the reddest circle on the main canvas.
Gradients: These are "unrolled" versions of the data. The Dynamic Gradient is a flattened representation of the colors of the connecting lines.

The Surprising Answer
So, which curves faster?
At the ends of the arc (near y=0 and y=π): The sine wave (cyan) curves much faster. Its curvature is at its maximum here, while the parabola is relatively straight.
In the middle of the arc (near y=π/2): The parabola (magenta) is now curving faster! The sine wave flattens out at its peak, and its curvature drops to its minimum.
The points where they swap dominance are the Crossover Points, marked with orange circles. These are the only two points where the curves are bending at the exact same rate.

The Big Picture: The Phase Diagram
The most powerful tool in the sidebar is the Crossover Phase Diagram. This chart answers the question: "How many crossover points will there be for any combination of sine amplitude and parabola height?"
Dark Blue (0 Crossovers): In this regime, the parabola is so "sharp" and the sine wave is so "flat" that the parabola is always curving more.
Orange (2 Crossovers): This is the standard case we've been looking at.
Yellow (4 Crossovers): In this rare and interesting state, the sine wave becomes so tall and "pointy" that it creates a second set of crossover points near its peak.