## The elementary differential geometry of plane curves |

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a-discriminant algebraic curve an+i angle centre of curvature changes sign chord circle of curvature concave convex conditions for contact contact of order continuous corresponding cusp defined definition differential coefficients direction cosines envelope evolute example following theorem follows at once geometrical given curve Goursat Hence implicit function theorem implicit functions instantaneous axis isolated characteristic points isolated points limit mean curvature moving axes necessary and sufficient neighbourhood neighbouring curves neighbouring point obtain order of smallness ordinary point osculating circle osculating curve osculating plane parabola parameters plane curve point of contact point of inflexion point Q points of intersection polynomial positive direction proof properties prove radius of curvature reader real root rectilinear asymptote referred Rolle's Theorem satisfy the equation second order contact shortest distance singular point spin straight line sufficient condition suppose surface tangent and normal Taylor's theorem twisted curve unique vanish