back to
Solid Geometry
Concepts of Three-dimensional Geometry
plane axiom 1
If two points of a line lie in a plane, all points of this line lie in the same plene. For example, A and B are two points on line l, if points A and B are on the plane P, all points on line l will be on plane P.
plane axiom 2
If there is a common point in two planes, the two plane has only one common line through this point. For example, P1 is a plane, P2 is another plane. A is a point in both plane, then there is only one common line l passing point A and lying on both plane P1 and P2.
plane axiom 3
Passing thought three points that are not in the same line has only a plane. For example, P is a plane, A, B, and C are three points which are not in the same line, passing thought three points A, B, and C makes only a plane P.
Plane inference 1
Passing thought one line and one point outside this line has only one plane. For example, P is a plane, m is a line passing thought points A and B. C is a point outside of line m, then passing thought point C and line m makes only a plane P.
Plane inference 2
Plane inference 3
Passing thought two parallel lines has only one plane. For example, P is a plane, m and n are two parallel lines, then passing thought m and n makes only one plane P.
Lines not in the same plane
Two lines are not in the same plane. For example, m is a line in left front of the cube, and n is another line in upper rear of the cube, since lines m and n lie different plane, m and n never intersect each other.
Line parallel to a plane
If there is no common point between a line and a plane, this line is parallel to the plane. For example, m is a line in left front of the cube, and P is a plane in the right side. Extending line m and plane P, m and P never intersect each other, then line m is parallel to plane P.
Line perpendicular to a plane
If a line m is perpendiculat to any lines in a plane P, the line m is perpendiculat to that plane P. For example, m is a line in upper front of the cube, P is a plane in right side of the cube. Since line m is perpendicular to line AC, AB, and AD, line m is perpendicular to plane P at point A.
Line perpendicular to a plane theory
If there are two lines perpendicular to a plane, these two lines are parallel. For example, m is a line in front left of the cube, n is a line in front right of the cube, P is a plane in bottom of the cube, since line m is perpendicular to plane P and line n is perpendicular to plane P, the line m parallel to line n.
Perpendicular point
Point B is the perpendicular point. (1). point B lies on the plane P (2). line AB is perpendicular to plane P.
Distance from a point to a plane
Draw a line passing through point A and perpendicular to the plane P, the distance between the point A and the perpendicular point B is the distance from point A to the plane P. For example, the distance from point A to the plane P is d = AB.
Distance from a line to a plane
If a line parallel to the plane, any distance drawing from this line to the plane is the distance between this line and the plane. For example, line m parallel to the plane P, the distance between the line m and the plane is d = d1 = d2.
Point of projection of a point outside of a plane to the plane
Draw a line passing thought this point and perpendiculat to a plane, the perpendicular point in this plane is called the point of projection. For example, A is a point outside of plane P, B is perpendicular point of A. Point B is on the plane P.
Perpendicular line of a point outside the plane to the plane
The distance between this point and the point of projection is called the perpendicular line of this point to the plane. For example, A is a point outside plane P, B is its point of projection, d is the perpendicular line of point A to the plane P, and d equals the length of AB.
Inclined line to a plane
A line intersects with a plane but not perpendicular to this plane. This line is called inclined line of this plane. For example, line EF intersects plane P at point F with an angle of a which is less than 90 degree, the line EF is called inclined line to the plane P.
Projection of an inclined line to a plane
Draw perpendicular lines from all points in the inclined line except F to plane P, connect all these points, this line is called projection line of the inclined line to this plane. For example, draw a line passing point E and perpendicular to plane P at point G, connect FG, the line FG is called projection of the inclined line EF to plane P.
Angle of an inclined line to a plane
An angle between an inclined line of a plane and its projection to this plane is called the angle of an inclined line to this plane. For example, a is the angle of inclined line EF to plane P. The range of a is: a is larger than 0 degree and a less than 90 degree.
Line perpendicular to a plane
If a line is perpendicular to two intersected lines in a plane, the line is perpendiculat to this plane. For example, line m and n intersected at point B in plane P. A is a point outside of plane P, if line AB is perpendicular to lines m and n, line AB is perpendicular to this plane P.
Lines perpendicular to a plane
If two line are parallel and both are not lie on a plane, if one of the lines is perpendicular to a plane, then the other line is also perpendicular to this plane. For example, lines m and n are parallel and both are not lie on plane P, if line m is perpendicular to plane P, then line n is also perpendicular to plane P.
Property of lines are perpendicular to a plane
If two lines are perpendicular to a plane at the same time, these two lines are parallel. For example, If lines m and n are perpendicular to plane P, then line m and line n are parallel.