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Let x be a point in an n-dimensional compact manifold M, and attach at x a copy of R^n tangential to M. The resulting structure is called the tangent space of M at x and is ...

The quartic surface given by the equation x^4+y^4+z^4-(x^2+y^2+z^2)=0.

The plane curve given by the equation xy=x^3-a^3, illustrated above for values of a ranging from 0 to 3. For a=0, the trident degenerated to a parabola.

The plane curve given by the equation axy = (a+x)(a-x)(2a-x) (1) = x^3-2ax^2-a^2x+2a^3, (2) which, solving for y, gives y=((a+x)(a-x)(2a-x))/(ax). (3) The plots above are for ...

cos(pi/(15)) = 1/8(sqrt(30+6sqrt(5))+sqrt(5)-1) (1) cos((2pi)/(15)) = 1/8(sqrt(30-6sqrt(5))+sqrt(5)+1) (2) cos((4pi)/(15)) = 1/8(sqrt(30+6sqrt(5))-sqrt(5)+1) (3) ...

A point where a curve intersects itself along three arcs. The above plot shows the triple point at the origin of the trifolium (x^2+y^2)^2+3x^2y-y^3=0.

To truncate a real number is to discard its noninteger part. Truncation of a (positive) number x therefore corresponds to taking the floor function |_x_|. Truncation also ...

A conjecture which treats the heights of points relative to a canonical class of a curve defined over the integers.

A linkage used in the original steam engine to turn back-and-forth motion into approximately straight-line motion.

If two curves of the same curve genus >1 are in rational correspondence, then that correspondence is birational.