Adding Points on an Elliptic Curve
The chord construction on the curve y² = x³ − 2x + 1: the line through two points meets the curve in a third point, and its mirror image is their sum.
An elliptic curve is the set of points with y^2 = x^3 + ax + b . Its points can be added, and the rule is a drawing. Take two points P and Q of the curve and draw the line through them. The line meets the curve in one more point, R (when the line touches the curve at P , R is P itself). The sum P + Q is the mirror image of R in the x -axis.
The third point has a formula. If the line has the slope m , putting its equation into the equation of the curve gives a cubic equation in x whose three roots add up to m^2 . Two of the roots are the x -coordinates s of P and k of Q , so the third is w = m^2 - s - k . With this addition, and one extra point at infinity that plays the part of zero, the points of the curve form a group: the order of the terms does not matter, (P + Q) + T = P + (Q + T) , and the mirror image of a point is its opposite. The cryptography that protects most connections on the web uses this group, on curves whose coordinates are whole numbers modulo a large prime.
This curve, y^2 = x^3 - 2x + 1 , has two parts, an oval and an open branch. P (blue) moves on the oval and Q (blue) on the branch; R is orange and P + Q is red. The document has no point that can be dragged, so the two sliders s and k set the x -coordinates of P and Q . Their ranges keep P and Q apart, so the line is never vertical. For some positions it touches the oval at P ; the orange point then covers P , and P + Q is the mirror image of P .
After a JSXGraph example by The Center of Mobile Learning with Digital Technology.
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