Tangents of the lower half of a circle

The bottom half of a circle has tangents that don't look anything like the tangents of a parabola or the trough of a sine wave. Most notably, the last two tangent lines, the one on the left and the one on the right, are both vertical.

Both the parabola and the circle have tangents that are neatly spaced out, but the parabola seen here only has slopes that range from -2 to 2, while the circle has slopes from negative infinity to infinity.

Tomorrow, a somewhat lesser known curve sometimes mistaken for a parabola, the catenary.

tangents of a sine wave trough

These are the tangent lines defined by the trough of a sine wave, with x between -pi and 0. Over this particular stretch, the curve might be hard to distinguish from the parabola from x varying from -1 to 1, but a close look at the tangent patterns should show how the two curves are different. Notice where the first downward sloping red tangent line meets the last upward sloping purple line.

Here in the sine wave curve, the first few tangents on the left are almost on top of each other, as are the last few tangents on the right.

Here are the parabola tangents again. The slopes of these tangents are increasing steadily, while the sine curve tangents max out at a slope of 1 and begin to get smaller as we leave the trough.

Tomorrow, another famous "round valley" curve, the bottom half of a circle.

Tangents of a parabola

I apologize to readers of this blog for my month's absence. My Internet provider was down for about eleven days and I prioritized the science fiction blog instead of this one.

Over the next few days, I'm going to play with describing a function using just the tangent lines. The first function is a parabola y = x² with x between -1 and 1 in increments of 0.1. Since this uses a total of 21 lines, I decided to use a seven colored rainbow sequence (ROY G. BIV for all you spectrum nerds out there) where three consecutive lines are red, then three orange, etc.
 

The second version of the idea still uses ROY G. BIV, but each line is a different color from the previous.

I think these are pretty and I'm going to try some other functions as well over the next few days.