An intersection is in mathematics, a common point of two curves in the plane or in space. The general parlance meant by the intersection of those two lines, but this is included in the mathematical curve concept. In space, there is the intersection of a curve with a surface. In the simplest case intersects with a straight one level.
The determination of an intersection point is in the two cases, line-line and line - level simple ( see below). Algemeinen in the determination of intersections of leads to non-linear equations to solve in practice, a Newton method. Intersections of a straight line with a conic section (circle, hyperbola, ellipse, parabola) or a quadric (sphere, ellipsoid, hyperboloid, ...) lead to quadratic equations and are still relatively easy to solve. For the intersection of a straight line with level / sphere / cylinder / cone provides descriptive geometry methods to determine intersections in the drawing.
- 2.1 Intersection of a line and a plane
- 2.2 intersection of three planes
- 2.3 points of intersection of a curve with a surface area
Intersection point in the plane
Intersection of two lines
For the intersection of two parallel non-
Obtained by Cramer's rule for the coordinates of the intersection point
(If the two lines are parallel. ) If the lines are given by two points: see next section.
Intersection of two lines
If two non-parallel lines and, so they do not overlap. Because the intersection of the corresponding lines must not be included in the two paths. To clarify the latter, it provides both routes is parameterized:
Intersect the lines, the common point of the associated line parameter must have with the property. The interface parameters are solution of the linear system
This is dissolved (as above) using the Cramer's rule checks the cutting condition, and sets a corresponding parameter or in the representation in order to finally to obtain the point of intersection.
Example: For the routes and obtained the system of equations
And. That is, the lines intersect and the intersection is.
Remark: If we consider the line through two pairs of points (not lines! ), So you can ignore the condition and obtained with this method the intersection of the two lines (see previous section ).
Intersections of a straight line with a circle
For editing, the
- Especially with the circle
Resulting from dissolution of the linear equation with respect to x or y, and insertion into the circle equation, the points of intersection with
If If true, there is only one intersection point and the straight line is a tangent to the circle.
Points of intersection of two circles
The determination of the points of intersection of two circles
Can be achieved by subtraction of the two equations, the problem points of intersection of the straight line
Traced back to one of the two circles (see above).
Intersections of two conics
The task of the intersection points of an ellipse / hyperbola / parabola with an ellipse / hyperbola / parabola to determine results in elimination of a coordinate ia to a quartic equation that is easily solved only in special cases. The intersections can be but also iteratively using the 1 - or 2- dimensional Newton's method to determine, depending on a) both conic sections implicitly ( -. > 2 -dim Newton ) or b) an implicit and the other is parameterized ( -> 1 -dim Newton ). . Please refer to the next section.
Intersection of two curves
Two in-plane, continuously differentiable curves ( ie curves without " kink" ) have an intersection if they have a point in the plane together and the two curves in this point either
If indeed the two curves have a common point, where a common tangent, but do not cross, they touch into.
Since touching cutting rarely occurs and is considered computationally very expensive, is always assumed transverse cutting below. To avoid having to mention it is not always provided, each necessary differentiability conditions are assumed. Determining intersections of leads again to the problem of having to solve an equation with one or two equations with two unknowns. The equations are i.a. non-linear and can usually with a 1 - 2 -dimensional Newton method are solved or. The individual cases and the equations to be solved are described below:
- If both curves explicitly present: provides equating the equation
- If both curves parameterized present.
- If a curve parameterized and, implicitly, the other are: .
- If both curves are given implicitly.
The necessary for the respective Newton method starting values can be obtained from a visualization of the two curves. A parameterized or explicitly given curve can be visualized easily, since one can directly compute a point x t to a given parameter or. For implicitly given curves this is not so simple. Here you have to i.a. calculated with the help of starting points and an iterative curve points.
Intersection of two polygons
If you are looking for intersections of two polygons, one can each individual stage of a polygon with each leg of the other polygon on cutting investigate (see above: the intersection of two lines ). This simple method is very time consuming for polygons with many sections. Through so-called window tests can significantly reduce the computation time. In this case, one considers a plurality of partial paths to a portion of the polygon and calculates the corresponding window, which is the minimum rectangle parallel to the axis that contains the part of polygon. Before consuming an intersection of two partial polygons is calculated, the associated window are tested for overlap.
Intersections in space
In three -dimensional space is called an intersection ( common point ) of a curve with an area. With the following considerations are ( as described above ), only the transverse sections of a curve with an area to be treated.
Intersection of a line and a plane
A straight line in space will be described generally by a parametric representation and a plane by an equation. Substituting the parametric representation of the resulting straight line in the plane equation linear direct drying
For the parameter of the intersection point. (If the linear equation has no solution, is the straight line parallel to the plane. )
Intersection of three planes
Is a two Just when cutting non-parallel planes given and is to be cut by a third plane, the common point of the three levels must be determined.
Three levels with linearly independent normal vectors have the intersection
To prove you convince yourself of in compliance with the rules for a scalar triple product.
Intersection points of a curve with a surface
Analogously as in the flat case, the following cases lead to ia non-linear systems of equations, with a 1 - can be solved or 3- dimensional Newton method.
- Parameterized curve and
- Parameterized curve and
Note: A straight line may also be contained in a plane. Then there are infinitely many points in common. Also, a curve can complete in an area be included (see curves on the surface ) partially or completely. In these cases, however, we no longer speak of the intersection.