This is a differential geometry lesson on parameterized plane curves, using the astroid (a four-cusped hypocycloid defined implicitly by x^(2/3)+y^(2/3)=1, or parametrically as r(t) = ⟨cos³t, sin³t⟩) to introduce vector-valued functions, the velocity vector (the derivative of the position vector, giving instantaneous direction and speed of motion along a curve), and the tangent line as the line through a curve point extending in the direction of the velocity vector via scalar multiplication (not a dot product). It establishes that a vector-valued function returns a single position vector at each parameter value rather than tracing the curve itself, and that cusps are points where the velocity vector vanishes, leaving the tangent line undefined.
This is a differential geometry lesson on parameterized plane curves, using the astroid (a four-cusped hypocycloid defined implicitly by x^(2/3)+y^(2/3)=1, or parametrically as r(t) = ⟨cos³t, sin³t⟩)…