Point on plane closest to origin
Encyclopedia
In Euclidean 3-space we will find the point
on an arbitrary plane
that is closest to the origin
using the method of Lagrange multipliers
.
First, let us start with an arbitrary plane, ax + by + cz = d. The distance, L, from the origin to a point (x,y,z) on the plane is given by:
Therefore the function
that we want to minimize is:
Our one constraint
on x, y, and z is that the point (x,y,z) must lie on the given plane. Thus, we define g(x,y,z) = ax + by + cz - d.
Next we define a new function with a Lagrange multiplier,
Take the partial of
with respect to x, y, and z and set each to zero.
Now each partial includes a
and a 
term.
If we solve each equation for
and set them equal to one another
we can find the relation:
From this we can obtain y and z as functions of x:
and
Substitute these for y and z in the equation
of the plane and solve for x to obtain:
With this x you can solve for y and z:
and
Hence the point on the plane closest to the origin is:
and the distance is given by:
Given a plane defined by three points
, 
, and 
.
The normal for this plane is
The nearest point
on the plane to the origin is the orthogonal projection of any point on the plane onto the plane's normal
Composed of the distance
from the plane to the origin, which is the dot product of 
and any point on the plane such as 
.
Point (geometry)
In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...
on an arbitrary plane
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...
that is closest to the origin
Origin (mathematics)
In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect...
using the method of Lagrange multipliers
Lagrange multipliers
In mathematical optimization, the method of Lagrange multipliers provides a strategy for finding the maxima and minima of a function subject to constraints.For instance , consider the optimization problem...
.
First, let us start with an arbitrary plane, ax + by + cz = d. The distance, L, from the origin to a point (x,y,z) on the plane is given by:
Therefore the function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
that we want to minimize is:
Our one constraint
Constraint (mathematics)
In mathematics, a constraint is a condition that a solution to an optimization problem must satisfy. There are two types of constraints: equality constraints and inequality constraints...
on x, y, and z is that the point (x,y,z) must lie on the given plane. Thus, we define g(x,y,z) = ax + by + cz - d.
Next we define a new function with a Lagrange multiplier,
Take the partial of
with respect to x, y, and z and set each to zero.Now each partial includes a
and a term.If we solve each equation for
and set them equal to one anotherwe can find the relation:
From this we can obtain y and z as functions of x:
and
Substitute these for y and z in the equation
Equation
An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...
of the plane and solve for x to obtain:
With this x you can solve for y and z:
and
Hence the point on the plane closest to the origin is:
and the distance is given by:
An applied solution to this problem using linear algebra
This approach is useful in computational geometry and applications of computer graphics.Given a plane defined by three points
, , and .The normal for this plane is
The nearest point
on the plane to the origin is the orthogonal projection of any point on the plane onto the plane's normalComposed of the distance
from the plane to the origin, which is the dot product of and any point on the plane such as .