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  2. Conjugate diameters - Wikipedia

    en.wikipedia.org/wiki/Conjugate_diameters

    For an ellipse, two diameters are conjugate if and only if the tangent line to the ellipse at an endpoint of one diameter is parallel to the other diameter. Each pair of conjugate diameters of an ellipse has a corresponding tangent parallelogram, sometimes called a bounding parallelogram (skewed compared to a bounding rectangle).

  3. Ellipse - Wikipedia

    en.wikipedia.org/wiki/Ellipse

    An ellipse (red) obtained as the intersection of a cone with an inclined plane. Ellipse: notations Ellipses: examples with increasing eccentricity. In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant.

  4. 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 ...

  5. Orthoptic (geometry) - Wikipedia

    en.wikipedia.org/wiki/Orthoptic_(geometry)

    If a tangent contains the point (x 0, y 0), off the parabola, then the equation = + = holds, which has two solutions m 1 and m 2 corresponding to the two tangents passing (x 0, y 0). The free term of a reduced quadratic equation is always the product of its solutions.

  6. Tangent - Wikipedia

    en.wikipedia.org/wiki/Tangent

    [1] [2] More precisely, a straight line is tangent to the curve y = f(x) at a point x = c if the line passes through the point (c, f(c)) on the curve and has slope f ' (c), where f ' is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space.

  7. Pedal curve - Wikipedia

    en.wikipedia.org/wiki/Pedal_curve

    Take P to be the origin. For a curve given by the equation F(x, y)=0, if the equation of the tangent line at R=(x 0, y 0) is written in the form ⁡ + ⁡ = then the vector (cos α, sin α) is parallel to the segment PX, and the length of PX, which is the distance from the tangent line to the origin, is p.

  8. Elliptic curve - Wikipedia

    en.wikipedia.org/wiki/Elliptic_curve

    Graphs of curves y 2 = x 3 − x and y 2 = x 3 − x + 1. Although the formal definition of an elliptic curve requires some background in algebraic geometry, it is possible to describe some features of elliptic curves over the real numbers using only introductory algebra and geometry.

  9. Parallelogram - Wikipedia

    en.wikipedia.org/wiki/Parallelogram

    For an ellipse, two diameters are said to be conjugate if and only if the tangent line to the ellipse at an endpoint of one diameter is parallel to the other diameter. Each pair of conjugate diameters of an ellipse has a corresponding tangent parallelogram , sometimes called a bounding parallelogram, formed by the tangent lines to the ellipse ...