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The vanishing point theorem is the principal theorem in the science of perspective. It says that the image in a picture plane π of a line L in space, not parallel to the picture, is determined by its intersection with π and its vanishing point. Some authors have used the phrase, "the image of a line includes its vanishing point".
Bregman–Minc inequality (discrete mathematics) Brianchon's theorem ; British flag theorem (Euclidean geometry) Brooks's theorem (graph theory) Brouwer fixed-point theorem ; Browder–Minty theorem (operator theory) Brown's representability theorem (homotopy theory) Bruck–Chowla–Ryser theorem (combinatorics) Brun's theorem (number theory)
The left-hand picture below is an example of a historic visual proof of the Pythagorean theorem in the case of the (3,4,5) triangle. Visual proof for the (3,4,5) triangle as in the Zhoubi Suanjing 500–200 BCE.
For example, a Fourier series of sine and cosine functions, all continuous, may converge pointwise to a discontinuous function such as a step function. Carmichael's totient function conjecture was stated as a theorem by Robert Daniel Carmichael in 1907, but in 1922 he pointed out that his proof was incomplete. As of 2016 the problem is still open.
Kawamata–Viehweg vanishing theorem; Kawasaki's Riemann–Roch formula; Keel–Mori theorem; Kempf vanishing theorem; Kempf–Ness theorem; Kodaira embedding theorem; Kodaira vanishing theorem; Krivine–Stengle Positivstellensatz
The point P is the vanishing point of lines perpendicular to the picture. If S is another point on the horizon, then it is the vanishing point for all lines parallel to OS . But Brook Taylor (1719) indicated that the horizon plane determined by O and the horizon was like any other plane :
The strange car-chase movie 'Vanishing Point' has had an equally strange afterlife, as detailed in this new book about the film and its star, a 1970 Dodge Challenger R/T 440.
Serre's vanishing theorem says that for any ample line bundle on a proper scheme over a Noetherian ring, and any coherent sheaf on , there is an integer such that for all , the sheaf is spanned by its global sections and has no cohomology in positive degrees.