# Many extraneous tangent lines

Exploring elliptic curves such as C = ImplicitCurve(x^3 - x - y^2). Then Tangent(C, P) for some point P on C, creates about 6 tangent lines, some of which appear to be duplicates. I desire the tangent line AT P, but the other lines given are tangents to C (passing through P), tangent at other points on C. The collection of lines is a single object, so I can't delete one or more lines without deleting all lines.

IF my curve were the graph of a function f, though it would be a pain since I am considering tangent lines at several points on C, I suspect I could define a new function, g, restricted to x-values within some small distance of the x-coordinate of my point P, and then construct Tangent(g, P). But I'm not sure if I can define a similar restriction of the implicit curve.

Better to hide the no desired lines

or you can do a custom tool

UPDATES: I only observe the above issue as follows. See attached file EllipticNxP.ggb .

(1) Begin with point X on the implicit curve eq1. Construct myTan2Eq1_X = Tangent(X, eq1), which works. Intersect eq1 & myTan2Eq1_X, which gave two points, say P & Q, where Q is exactly at the location of X. Construct Tangent(P, eq1), and it is here where it generates five (5) tangent lines [labelled T_1, ..., T_5; colored gray], not just the true local tangent line.

(2) Oh, I tried a thought above in my original post, of restricting the implicit curve:

eq2 = ImplicitCurve((x^3 - x - y^2)/((x - x(F)<epsilon)*(y - y(F)<epsilon)))

This indeed does graph the restricted implicit curve, eq2. BUT SOMETIMES I cannot intersect this restricted curve with another path object, even when I tried setting epsilon = 50, thus obtaining a "complete" picture of the curve.

I constructed a free line h = Line(G, H), Intersecting line h with the restricted eq2, I was able to obtain 3 intersection points, R, S, T on eq2. Trying to construct myTan2Eq2_R = Tangent(R, eq2) produced nothing whatsoever-- no tangent line, no entry in the Algebra listing of constructed objects. On the other hand myTan2Eq1_S = Tangent(S, eq1) did produce 5 tangent lines at S [colored yellow]. Further, I also intersected line h with the unrestricted eq1, producing points I, J, K, with I at the location of R, J at the location of S, K at the location of T. Constructing myTan2Eq1_K produced a single tangent line at K. Attempting to construct myTan2Eq2_I produced nothing whatsoever.

El manejo de curvas implícitas tiene los mismos o más requisitos que el manejo de ecuaciones, a veces no es posible manejar todas las ecuaciones luego no es posible manejar todas las curvas implícitas.

Es conveniente no complicar las expresiones de dichas curvas y sobre todo no usar expresiones no derivables o no continuas en ellas. por ejemplo myTan2Eq2_R = Tangent(R, eq2) no es manjable porque la expresion de eq2 no es continua por las comparaciones que tiene su expresion. pero definiendo eq3=implicitcurve(f(x)-y^2) entonces myTan2Eq3_R = Tangent(R, eq3) sí es obtenida. se puede entonces ver si las condiciones se cumplen o no y descartarla segun el resultado.

Un ejemplo simple de error en el cálculo de la tangente

More ImplicitCurve trouble: With Neg(x) = - x, f(x) = (x+1)(x)(x-1), and Gold = {1+\sqrt{5})/2, then Neg intersects elliptic curve eq1 = ImplicitCurve(f(x) - y^2) at three points: A = (1- Gold, Gold}-1), B = (0,0), C = (Gold}, - Gold). Create X = Point(eq1). Then SetValue(X, A) and SetValue(X, Point({1 - Gold, Gold - 1})) only place X near the correct location. Repeating the SetValue commands does not push X any closer to the correct coordinates.

How can I get X to move to the correct location on eq1 ?

See attached file.

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