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  2. Tangent lines to circles - Wikipedia

    en.wikipedia.org/wiki/Tangent_lines_to_circles

    If = + is the distance from c 1 to c 2 we can normalize by =, =, = to simplify equation (1), resulting in the following system of equations: + =, + =; solve these to get two solutions (k = ±1) for the two external tangent lines: = = + = (+) Geometrically this corresponds to computing the angle formed by the tangent lines and the line of ...

  3. Contact (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Contact_(mathematics)

    A circle with 1st-order contact (tangent) A circle with 2nd-order contact (osculating) A circle with 3rd-order contact at a vertex of a curve. For each point S(t) on a smooth plane curve S, there is exactly one osculating circle, whose radius is the reciprocal of κ(t), the curvature of S at t.

  4. Tangent circles - Wikipedia

    en.wikipedia.org/wiki/Tangent_circles

    In geometry, tangent circles (also known as kissing circles) are circles in a common plane that intersect in a single point. There are two types of tangency : internal and external. Many problems and constructions in geometry are related to tangent circles; such problems often have real-life applications such as trilateration and maximizing the ...

  5. Dual curve - Wikipedia

    en.wikipedia.org/wiki/Dual_curve

    Let Xx + Yy + Zz = 0 be the equation of a line, with (X, Y, Z) being designated its line coordinates in a dual projective plane. The condition that the line is tangent to the curve can be expressed in the form F(X, Y, Z) = 0 which is the tangential equation of the curve. At a point (p, q, r) on the curve, the tangent is given by

  6. Tangent - Wikipedia

    en.wikipedia.org/wiki/Tangent

    Tangent to a curve. The red line is tangential to the curve at the point marked by a red dot. Tangent plane to a sphere. In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point.

  7. Problem of Apollonius - Wikipedia

    en.wikipedia.org/wiki/Problem_of_Apollonius

    The property of tangency is defined as follows. First, a point, line or circle is assumed to be tangent to itself; hence, if a given circle is already tangent to the other two given objects, it is counted as a solution to Apollonius' problem. Two distinct geometrical objects are said to intersect if they have a point in common. By definition, a ...

  8. Curvature - Wikipedia

    en.wikipedia.org/wiki/Curvature

    is equal to one. This parametrization gives the same value for the curvature, as it amounts to division by r 3 in both the numerator and the denominator in the preceding formula. The same circle can also be defined by the implicit equation F(x, y) = 0 with F(x, y) = x 2 + y 2 – r 2. Then, the formula for the curvature in this case gives

  9. Ford circle - Wikipedia

    en.wikipedia.org/wiki/Ford_circle

    In mathematics, a Ford circle is a circle in the Euclidean plane, in a family of circles that are all tangent to the -axis at rational points. For each rational number p / q {\displaystyle p/q} , expressed in lowest terms, there is a Ford circle whose center is at the point ( p / q , 1 / ( 2 q 2 ) ) {\displaystyle (p/q,1/(2q^{2}))} and whose ...