Plane (geometry)

Graphics of two hyperboles and their asymptotes on the Cartesian plane.

Informal graphical representation of a plane.

In geometry, a plane is an ideal object that only has two dimensions, and contains infinitely many points and lines; It is a fundamental concept of geometry along with the point and the line.

When talking about a polyline plane, we are talking about a geometric object that has no volume, that is, two-dimensional, and that contains an infinite number of lines and points. However, when the term is used in the plural, one is talking about that object elaborated as a graphic representation of surfaces in different positions. Plans are especially used in engineering, architecture and design, since they are used to diagram on a flat surface or on other surfaces that are regularly three-dimensional.

A plane is defined by the following geometric elements:

• Three unaligned points.
• A straight and an outer point to it.

• Two parallel or two straight lines that are cut.

Planes are often named after a letter of the Greek alphabet.

It is usually represented graphically, for better visualization, as a figure delimited by irregular edges (to indicate that the drawing is a part of an infinite surface).

In a Cartesian coordinate system, a point in the plane is determined by an ordered pair, called the abscissa and ordinate of the point. Through this procedure, two ordered real numbers (abscissa and ordinate) always correspond to every point in the plane, and reciprocally, a single point in the plane corresponds to an ordered pair of numbers. Consequently, the Cartesian system establishes a one-to-one correspondence between a geometric concept such as points in the plane and an algebraic concept such as ordered pairs of numbers. In polar coordinates, by an angle and a distance. This correspondence constitutes the foundation of analytic geometry.

The area is a measure of extension of a surface, or of a flat geometric figure, expressed in units of measure called surface units. For flat surfaces the concept is more intuitive. Any flat surface with straight sides, such as a polygon, can be triangulated and its area can be calculated as the sum of the areas of these triangles. Occasionally the term "area" as a synonym of surface, when there is no confusion between the geometric concept itself (surface) and the metric magnitude associated with the geometric concept (area).

Properties of plane ℝ3

Two-plan intersection in a three-dimensional space. Isometric representation of two perpendicular planes.

In a three-dimensional Euclidean space ℝ³, we can find the following facts (which are not necessarily valid for higher dimensions):

• Either two planes are parallel, or intersect in one line.
• Either a straight is parallel to a plane, or it is intersected with it at a point, or it is contained in it.
• Two straights perpendicular to the same plane are parallel to each other.
• Two planes perpendicular to the same line are parallel to each other.
• Between a plane κ either and a straight not perpendicular to it there is only one plane such that it contains to the straight and is perpendicular to the plane.
• Between a plane κ either and a straight perpendicular to the same there are infinite planes such that contain the straight and are perpendicular to the plane.

Vector equation of the plane

A plane is defined by the following geometric elements: a point and two vectors:

Point P = (x₁, y₁, z₁)
Vector u = (u_x, u_y, u_z)
Vector v = (a₂, b₂, c₂)

(x,and,z)=(x1,and1,z1)+m(ux,uand,uz)+n(a2,b2,c2){displaystyle (x,y,z)=(x_{1},y_{1},z_{1})+m(u_{x},u_{y},u_{z})+n(a_{2},b_{2},c_{2}),!}

where m{displaystyle m} and n{displaystyle n} They're climbers.

This is the vector form of the plane; however, the most used form is the reduced one, the result of equaling to zero the determinant formed by the two vectors and the generic point X = (x, y, z) with the given point. Thus, the equation of the plane is:

{begin{vmatrix}x-P_{x}&y-P_{y}&z-P_{z}\u_{x}&u_{y}&u_{z}\v_{x}&v_{y}&v_{z}end{vmatrix}}=0=>Ax+By+Cz+D=0}" xmlns="http://www.w3.org/1998/Math/MathML">日本語(X− − P)uv日本語=0= 2005日本語x− − Pxand− − Pandz− − Pzuxuanduzvxvandvz日本語=0= 2005Ax+Band+Cz+D=0{cHFFFFFF}{cHFFFF}{cHFFFF}{cHFFFF}{cHFF}{cHFFFF}{cHFFFF}{v}}{vmatrix}}}}{x1⁄2}{vmcHFFFF}{v}{v}{x1⁄4}{x1}{x1⁄4}{x1⁄4}{x}{x1⁄4}}{x1⁄4}{x1⁄4}{x1⁄4}}{x1⁄4}{x1⁄4}}}}}{x1⁄4}{x1⁄4}{x1⁄4}{x1⁄4}}{x1⁄4}}{x1⁄4}}{x1⁄4}{x1⁄4}}{x1⁄4}}{x1⁄4}}{x1⁄4}{x1⁄4}}{x1⁄4}}{x1⁄4}}}{x1⁄4}}}}}}{{begin{vmatrix}x-P_{x}&y-P_{y}&z-P_{z}\u_{x}&u_{y}&u_{z}\v_{x}&v_{y}&v_{z}end{vmatrix}}=0=>Ax+By+Cz+D=0" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/104ae0dfed89f55e4bae7eb3d9276aae76eafa05" style="vertical-align: -4.338ex; width:78.914ex; height:9.843ex;"/>

Where (A, B, C) is a vector perpendicular to the plane and coincides with the vector product of the vectors u and v. The formula to find the equation when it is not at the origin is:

a(x− − h)+b(and− − k)+c(z− − j)=0{displaystyle a(x-h)+b(y-k)+c(z-j)=0,}

Strictly

P = P₀ + mA + nB is the equation of the plane determined by a fixed point and two non-collinear vectors A and B.

Equation by orthogonal vector

a.x = 0, where a is an orthogonal vector and x is a point in the plane.

Relative position between two planes

If we have a plane 1 with a point A and a normal vector 1, and we also have a plane 2 with a point B and a normal vector 2.

Their relative positions can be:

• Matching plans: the same address of normal vectors and point A belongs to plan 2.
• Parallel plans: if they have the same direction normal vectors and point A does not belong to plan 2.
• Dry plans: if normal vectors do not have the same direction.

Distance from a point to a plane

For any plane Русский Русский :ax+band+cz+d=0{displaystyle Pi:ax+by+cz+d=0,} and a point p1=(x1,and1,z1){displaystyle mathbf {p} _{1}=(x_{1},y_{1},z_{1})}}} not necessarily contained in that plane Русский, the lower distance between P₁ and plan Русский is:

D=日本語ax1+band1+cz1+d日本語a2+b2+c2.{displaystyle D={frac {left entailax_{1}+by_{1}+cz_{1}+dright responsible}{sqrt {a^{2}+b^{2}+c^{2}}}}}}}}}}}}{. !

From the above it follows that the point P₁ will belong to the plane Π if and only if D= 0.

If the coefficients a, b and c of the canonical equation of any plane are normalized, that is when a2+b2+c2=1{displaystyle {sqrt {a^{2}+b^{2}+c^{2}=1}, then the previous formula of distance D is reduced to:

D=日本語ax1+band1+cz1+d日本語.{displaystyle D= Δax_{1}+by_{1}+cz_{1}+dą. !

Half-flat

Square plan.

In geometry, each of the two parts into which a plane is divided by a line is called a half-plane.

Analytical

Inecution ax+band+c≥ ≥ 0{displaystyle ax+by+cgeq 0} determines a semi-plane and its straight border ax+band+c=0{displaystyle ax+by+c=0}

Inecution 0}" xmlns="http://www.w3.org/1998/Math/MathML">ax+band+c▪0{displaystyle ax+by+c 2005}0}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f5302ad1f697235530aae1d96994cfd3badd772" style="vertical-align: -0.671ex; width:15.661ex; height:2.509ex;"/> determines a semi-plane without including the border ax+band+c=0{displaystyle ax+by+c=0}. This semi-plane is a convex, open and non-accused set.

Partition

The equation line L=ax+band+c=0{displaystyle L=ax+by+c=0}and the semi-planes <math alttext="{displaystyle S_{1}=ax+by+cS1=ax+band+c.0{displaystyle S_{1}=ax+by+c excl0}<img alt="{displaystyle S_{1}=ax+by+c, 0}" xmlns="http://www.w3.org/1998/Math/MathML">S2=ax+band+c▪0{displaystyle S_{2}=ax+by+c/2005}0}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/577df3a9d1f07fe0d4e18dec9ebc027bbbb63e76" style="vertical-align: -0.671ex; width:21.239ex; height:2.509ex;"/> determine a partition of the plane, so that any point of this is exactly in one, and only one of the three sets: straight LSemi-planes S1{displaystyle S_{1}} or S2{displaystyle S_{2}}.

Postulates of the division of a plane

In each pair of half planes that a line r determines on a plane there are infinitely many points such that:

1. Every point of the plane belongs to one of the two semi-planes or to the straight that determines them.
2. Two points of the same semiplane determine a segment that does not cut to the straight r.
3. Two different semiplane points determine a segment that cuts to the straight 'r8.