Maths

The tangent equations for circles, ellipses, and hyperbolas share a noticeable pattern: replace one factor in each squared term with the corresponding coordinate of the point of contact, and the quadratic equation becomes a tangent line. Why does this work? How should we handle the linear term in a parabola, or the mixed term in a general quadratic curve? I want to start with a circle and work out the rule step by step. ...

When working on an ellipse problem, I sometimes pause to ask: would this become simpler if I compressed the ellipse into a circle? A circle has familiar geometric relationships between tangents, radii, and chord midpoints. With a suitable change of coordinates, those relationships can also help us solve problems about ellipses. ...

High-school physics tells us that the displacement in simple harmonic motion (SHM) is a sinusoidal function. But textbooks never explain why — they assume a solution of the form $x(t)=\sin(\cdots)$, substitute it into the equation, verify that it works, and call it a day. That is a working backwards from the unknown to the known approach, and it has always left me uneasy. So I decided to do the opposite: starting from Hooke’s law as the sole axiom, I would analyse the evolution of velocity step by step and see whether I could derive the unknown from the known — letting that sine (or cosine) emerge on its own. This post records my two attempts: the first using brute-force infinite subdivision, the second using matrices and geometry. From complicated to clean, I ended up colliding head-on with Euler’s formula. ...