What does the curve used in Bitcoin, secp256k1, look like?
I'm afraid you won't like the answer.

These curves - including the secp256k1 curve, y2 = x3 + 7` - 'look' nice when evaluated in typical number fields (integers, reals, ...), but secp256k1 is defined over the field Z2256-232-977, which means the X and Y coordinates are 256-bit integers modulo a large number. Curves using such coordinates do not have nice continuity properties.

I've tried to plot this curve over a similar but much smaller field, Z28+1. Coordinates extend from -128 to 128.



Note that even though it may not make sense geometrically anymore, it still has all properties you need. A line (which means, a group of points with equation ay + bx + c = 0) that 'intersects' 2 points of the curve, will intersect a third. Tangent again has no geometric interpretation anymore, but you can still compute a local linear approximation for the curve equation in a given point, which will have the property of intersecting the curve in a second point.

To show you what you'd get if this were over the real numbers, here is a plot of the same curve equation for that case. Once with coordinates -128 through 128, once with -8 through 8.