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  2. Intersection (geometry) - Wikipedia

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

    one solves the line equation for x or y and substitutes it into the equation of the circle and gets for the solution (using the formula of a quadratic equation) (,), (,) with x 1 / 2 = a c ± b r 2 ( a 2 + b 2 ) − c 2 a 2 + b 2 , {\displaystyle x_{1/2}={\frac {ac\pm b{\sqrt {r^{2}(a^{2}+b^{2})-c^{2}}}}{a^{2}+b^{2}}}\ ,}

  3. Steinmetz solid - Wikipedia

    en.wikipedia.org/wiki/Steinmetz_solid

    The generation of a bicylinder Calculating the volume of a bicylinder. A bicylinder generated by two cylinders with radius r has the volume =, and the surface area [1] [6] =.. The upper half of a bicylinder is the square case of a domical vault, a dome-shaped solid based on any convex polygon whose cross-sections are similar copies of the polygon, and analogous formulas calculating the volume ...

  4. Spherical cap - Wikipedia

    en.wikipedia.org/wiki/Spherical_cap

    For example, assuming the Earth is a sphere of radius 6371 km, the surface area of the arctic (north of the Arctic Circle, at latitude 66.56° as of August 2016 [7]) is 2π ⋅ 6371 2 | sin 90° − sin 66.56° | = 21.04 million km 2 (8.12 million sq mi), or 0.5 ⋅ | sin 90° − sin 66.56° | = 4.125% of the total surface area of the Earth ...

  5. Intersecting chords theorem - Wikipedia

    en.wikipedia.org/wiki/Intersecting_chords_theorem

    The value of the two products in the chord theorem depends only on the distance of the intersection point S from the circle's center and is called the absolute value of the power of S; more precisely, it can be stated that: | | | | = | | | | = where r is the radius of the circle, and d is the distance between the center of the circle and the ...

  6. Line–line intersection - Wikipedia

    en.wikipedia.org/wiki/Line–line_intersection

    In order to find the intersection point of a set of lines, we calculate the point with minimum distance to them. Each line is defined by an origin a i and a unit direction vector n̂ i. The square of the distance from a point p to one of the lines is given from Pythagoras:

  7. Line-cylinder intersection - Wikipedia

    en.wikipedia.org/wiki/Line-cylinder_intersection

    Cyan line has a single point of intersection. Green line has two intersections. Yellow line lies tangent to the cylinder, so has infinitely many points of intersection. Line-cylinder intersection is the calculation of any points of intersection, given an analytic geometry description of a line and a cylinder in 3d space.

  8. Intersection curve - Wikipedia

    en.wikipedia.org/wiki/Intersection_curve

    The analytic determination of the intersection curve of two surfaces is easy only in simple cases; for example: a) the intersection of two planes, b) plane section of a quadric (sphere, cylinder, cone, etc.), c) intersection of two quadrics in special cases. For the general case, literature provides algorithms, in order to calculate points of ...

  9. Ellipsoid - Wikipedia

    en.wikipedia.org/wiki/Ellipsoid

    Given: Ellipsoid ⁠ x 2 / a 2 ⁠ + ⁠ y 2 / b 2 ⁠ + ⁠ z 2 / c 2 ⁠ = 1 and the plane with equation n x x + n y y + n z z = d, which have an ellipse in common. Wanted: Three vectors f 0 (center) and f 1, f 2 (conjugate vectors), such that the ellipse can be represented by the parametric equation