Centroid

The geometric center of gravity or centroid of a geometrical figure (for example, arc, triangle, cone) is a particularly excellent point that you interpret for unsymmetrical figures as a kind of center. Mathematically, this corresponds to the average of all points within the figure. Specifically, the geometric center of gravity of lines and line centroid centroid of surfaces and volumes of bodies focus is called. The focus can be obtained in simple cases by geometrical considerations, or generally calculated with means of mathematics through integration. To describe the body's methods of analytical geometry are used. The focus is a Gravizentrum.

The geometric center of gravity corresponds to the center of mass of a physical body, which consists of a homogeneous material, so all have the same density. It can therefore be determined by purely mechanical balancing. This method can be applied to models when it comes to geographical centers of continents or countries (for example the center of Europe or the center of Germany ).

• 3.1 lines
• 3.2 surfaces
• 3.3 body
• 3.4 General

• 5.1 line center of a circular arc
• 5.2 centroid of a parabola

Priorities of elementary geometric figures

Below are some priorities elementary geometric lines, surfaces and solids are given and partly motivated by geometric considerations.

For axisymmetric or rotationally symmetrical figures, the disclosure of gravity simplified by the fact that this is always on the axis of symmetry. For figures with multiple axes of symmetry or point- symmetric objects, such as in a square or a circle, the focus is the intersection of the axes of symmetry (center ) of the figure.

Lines

Straight Line

The focus of a straight line of the length is in their midst:

Arc

If the section of the circle is rotated and shifted so that the y- axis of the Cartesian coordinate system is a symmetry axis of the arc and the center of the circle at the origin is (see picture), then can the focus by

Calculate. Here, the radius of the circle, the length of the arc and the chord length of the arc.

For the formula fails. With the center of gravity can be calculated even for very small angles.

Had to be the first district to be moved or rotated, then it must be moved back or rotated according again to complete the account of the calculated center of gravity.

Flat sheet

To calculate the centroid of a flat sheet is approximately, it must be shifted in the Cartesian coordinate system so that the center of the line connecting the two end points located at the origin. Then the center of gravity is a good approximation for slightly below

In ( semi-circle), the focus is exactly when. The percentage deviation increases roughly proportional to h and is approximately 4.7%. It follows from the expression that specifies the center of gravity in the range of with an accuracy of better than 5 parts per thousand. The exact location of the line of gravity in the whole range of can be found by substituting in the formula for the related to the circle center point of gravity ( see upper section arc ):

Interestingly, shows a maximum slightly larger than. War at the beginning of a shift or rotation needed, so the focus must be shifted back again accordingly.

Level surfaces

In flat areas, the focus can be generally characterized determine that you hang the cut surface at a point and the Perpendicular, called a gravity line inscribes. The intersection of two lines is the gravity center of gravity. All other Heavy lines intersect also in this priority.

Triangle

The medians of a triangle are gravity lines of the triangle. His focus is in the common intersection of the three medians. He shares this in a 2:1 ratio with the longer of the two distances is the distance from the centroid to the vertex.

Are the Cartesian coordinates of the vertices of the triangle are known, then gives the focus as the arithmetic mean.

Its barycentric coordinates are therefore.

In terms of trilinear coordinates is the center of gravity of a triangle with side lengths,

You can also focus using the length of one side and the height above the same side determine in Cartesian coordinates. The origin of the coordinate system is located in the corner ( see illustration). In this way, the Cartesian coordinates of the center of gravity can be carried

Calculate.

Trapeze

The focus of the trapezoid can be constructed as follows: A centroid line bisects the two parallel sides. A second is obtained by the parallel sides of the other is extended by the length in the opposite directions, and the two end points with each other. The formula in Cartesian coordinates is ( measured from the bottom left corner ):

Polygon

The center of gravity of a non- depressed, closed, and irregular polygon with N vertices can be calculated from the Cartesian coordinates of the vertices ( the zeroth corner (x0, y0 ) and the Nth vertex ( Xn, Yn) are in this case identical) as follows. The focus of a regular polygon corresponds to the center of its circumcircle.

The area of ​​the polygon can with the Gaussian triangle formula

Be determined. The centroid of the polygon is then represented by the formulas

Determined.

Circular cutout

If the section of the circle is rotated and shifted so that the y- axis of the Cartesian coordinate system is a symmetry axis of the sector and the center ( the full circle ) at the origin is (see picture), then center of gravity can by in radians

With calculated.

For the formula fails. With the approximation: the emphasis can also be calculated.

Had to be the circle initially moved or rotated, then it must be moved back or rotated according again to complete the account of the calculated center of gravity.

Circular section

To calculate the surface center of gravity of a circular section approximately, it must be shifted in the Cartesian coordinate system so that the center of the line connecting the two end points located at the origin. Then the center of gravity is a good approximation for a little above

In ( semi-circle), the focus is exactly at. The percentage deviation increases roughly proportional to h and is at about 5.8%. It follows from the expression that specifies the center of gravity in the range of with an accuracy of better than 5 parts per thousand. The exact location of the centroid in the entire range of can be found by substituting in the formula for the related to the circle center point focus:

War at the beginning of a shift or rotation needed, so the focus must be shifted back again accordingly.

Body

Three-dimensional body can be both the volumetric center of gravity, so the center of gravity of the solid body, and the centroid, that is, the center of gravity of the surface defining the body, calculate.

Pyramid and cone

To calculate the volume center of gravity and the center of gravity of a pyramid or a cone, it moves in the oblique coordinate system, so that the center of gravity of the base is located at the origin, and the Y axis passes through the apex. Then, by the volumetric center of gravity of a pyramid or a cone

And the surface center of gravity of the lateral surface by

Be calculated.

Paraboloid of revolution

To calculate the volume center of gravity and the center of gravity of a paraboloid of revolution, it is moved in the Cartesian coordinate system so that the center of gravity of the base is located at the origin. Then you can focus the volume of the paraboloid of revolution by

Calculate. The centroid looks a little more complicated. For the components and also applies again

And the component is

Wherein said expression is in the denominator of the first fracture, the circumferential surface of the open right parabola with the focal length f. From tends to, otherwise against.

Spherical segment

To calculate the volume center of gravity and the center of gravity of a spherical segment, shifts to the segment in the Cartesian coordinate system so that the center of the solid sphere is located at the origin. The volume focus is then

And the centroid by

Calculated. ()

Summarizing main points

It is possible to combine several areas of individual characters to a common center of gravity of the total figure, so that the result of the gravity of a composite figure from the priorities of individual simple elements.

The coordinates, and must be expressed in an arbitrary but uniform Cartesian coordinate system. Assigns an area ( a body ) recesses, so the above empirical formulas can also be applied taking into account that the recessed surfaces ( volume) in the calculation with a negative sign. The components of the center of gravity form the vector.

Definition of center of gravity by integrals

The formulas for calculating the center of gravity elementary geometric figures can be derived with the integrals given below. For more complicated figures, these integrals can often only be determined numerically.

The definition corresponds mathematically to the average of all points of the geometric object ( the body) in the Euclidean space. In lines and spaces in two-dimensional space, only the coordinates and to calculate the coordinate is omitted. The integration range is one-dimensional in lines, two-dimensional surfaces and in three-dimensional bodies.

Lines

For a line the length of the center of gravity is given by

With

These integrals are line integrals of the first kind

Surfaces

For a surface with surface area of ​​the center of gravity is defined by

With

These integrals are surface integrals with scalar surface element.

Body

In the case of a bounded body in three-dimensional space with volume, the focus is defined by

With

These integrals are volume integrals.

Generally

Is a body with the volume. The focus of is defined by

Where the m-dimensional volume element and the dimension of that is.

Integration with symmetric objects

For objects possess the symmetry elements, such as an axis of symmetry or a plane of symmetry, the calculation of the centroid simplified in many cases, as the focus is always included in the symmetry element. If the object has an axis of symmetry, the volume element can be expressed as a function of the infinitesimal element axis. So it only needs to be integrated over the axis of symmetry.

Examples of integral calculus

Line center of a circular arc

Points on a plane circular arc can be most easily specified in polar coordinates. If the y-axis lies on the symmetry line originating in the center of the circle, the coordinates are:

The length of the arc is given by:

Where the infinitesimal length element can be substituted by.

For reasons of symmetry. For the y- coordinate of the line center arises from the defining equation:

The integration within the limits then yields

Centroid of a parabola

For quick determination of the x-coordinate of the centroid in the 2 -dimensional case substituted by one, which corresponds to an infinitesimal surface strips. Furthermore, this corresponds to the surface bounding function.

For the practical calculation of the y- coordinate in the two -dimensional case, there are basically two approaches:

• Either by forming inverse function and calculates the integral, with the " new" limits of integration can now be found on the y -axis.
• Or one utilizes the fact that the center of gravity of each parallel to the y- axis infinitesimal surface strip. Then you get to determine the y-coordinate of a simpler formula with the help of the form of the inverse function are spared:

We are looking for the centroid of that area, which is defined by a parabola and the x - axis ( see figure ).

First we determine the content of the surface

The limits of the integrals are the zeros of the function for limiting the area of the x-axis.

The coordinate of the center of gravity is given by

The coordinate is given by

Another way, the center of gravity coordinates obtained to calculate an area characterized by the formulas

Wherein the borders and the intersection of the functions and displayed. This formula can be the focus of any flat surface which is enclosed between two functions that calculate. Conditions for this are