Intersections of Lines, Segments and Planes (2D & 3D)
A point where three or more lines intersect is called a point of concurrency. of a triangle is a straight line that divides the angle into two congruent angles. Two or more lines which share exactly one common point are called The intersecting lines meet at one, and only one point, no matter at what angle they meet. If a pair of lines lie in the same plane and do not intersect when produced on A straight line which cuts two or more straight lines at distinct points is called a.
It also has three vertices, which are each corner where two edges meet.
Parallel & perpendicular lines intro
Sciencing Video Vault You can also see from this definition that some two-dimensional shapes do not have any vertices. For example, circles and ovals are made from a single edge with no corners.How to find the intersection point of two linear equations
Since there are no separate edges intersecting, these shapes have no vertices. A semi-circle also has no vertices, because the intersections on the semi-circle are between a curved line and a straight line, instead of two straight lines.
Vertices of Three-Dimensional Shapes Vertices are also used to describe points in three-dimensional objects. Three-dimensional objects are composed of three different parts. Each line where two faces meet is called an edge.
Each point where two or more edges meet is a vertex.
Parallel & perpendicular lines | Basic geometry (video) | Khan Academy
A cube has six square faces, twelve straight edges, and eight vertices where three edges meet. In other words, each of the cube's corners is a vertex.
As with two-dimensional objects, some three-dimensional objects -- such as spheres -- do not have any vertices because they do not have intersecting edges.
Vertex of a Parabola Vertices are also used in algebra. A parabola is a graph of an equation that looks like a giant letter "U. It is shown there how to convert from other representations to the parametric one. In any dimension, the parametric equation of a line defined by two points P0 and P1 can be represented as: Using this representation, and whenP s is a point on the finite segment P0P1 where s is the fraction of P s 's distance along the segment.
Let two lines be given by: For andthis means that all ratios have the value a, or that for all i.
This is equivalent to the conditions that all. In 2D, withthis is the perp product [Hill, ] condition that wherethe perp operator, is perpendicular to u. This condition says that two vectors in the Euclidean plane are parallel when they are both perpendicular to the same direction vector. When true, the two associated lines are either coincident or do not intersect at all. Coincidence is easily checked by testing if a point on one line, say P0, also lies on the other line Q t.
That is, there exists a number t0 such that: In 2D, this is another perp product condition: If this condition holds, one hasand the infinite lines are coincident. And if one line but not the other is a finite segment, then it is the coincident intersection.
However, if both lines are finite segments, then they may or may not overlap. If the segment intervals [t0,t1] and [0,1] are disjoint, there is no intersection. Otherwise, intersect the intervals using max and min operations to get. Then the intersection segment is. This works in any dimension.
- Geometric definitions example
- Lines, Rays, and Angles
- Intersecting Lines And Non-intersecting Lines
Non-Parallel Lines When the two lines or segments are not parallel, they might intersect in a unique point. In 2D Euclidean space, infinite lines always intersect.
In higher dimensions they usually miss each other and do not intersect. But if they intersect, then their linear projections onto a 2D plane will also intersect. To compute the 2D intersection point, consider the two lines and the associated vectors in the diagram: To determine sI, we have the vector equality where. At the intersection, the vector is perpendicular toand this is equivalent to the perp product condition that.
Solving this equation, we get: Note that the denominator only when the lines are parallel as previously discussed. Similarly, solving for Q tIwe get: The denominators are the same up to sign, sinceand should only be computed once if we want to know both sI and tI. Further, if one of the two lines is a finite segment or a raysay P0P1, then the intersect point is in the segment only when or for a ray. If both lines are segments, then both solution parameters, sI and tI, must be in the [0,1] interval for the segments to intersect.
Plane Intersections Planes are represented as described in Algorithm 4, see Planes. Let L be given by the parametric equation: