The Reuleaux triangle or Equilateral Triangle

The coordinates of mind, P, satisfy the relation which is undercoat by eliminating the ratios -1:(:(. These coordinates argon as follows:

?x x1 x2? = 0.

(y1z2 - y2z1)x + (z1x2 - z2x1)y + (x1y2 - x2y1)z = 0

This equation, being of the progress to lx + my + nz = 0, is called a linear equation.

Every linear equation determines a line. Moreover, each such equation determines a line uniquely. For example, the given equation lx + my + nz = 0 describes coordinates of points Q(-n,0,l) and R(m,-l,0). This gives the equation of the corking line QR as follows:

? x y z? = 0 or l(lx + my + nz) = 0.

Furthermore, this line is unique. The two points Q(-n,0,l) and R(m,-l,0) are the only points on QR whose y, z coordinates are nix, respectively.

Two straight lines having one commons point can be draw by the succeeding(a) two equations:

l1x + m1y + n1z = 0 and l2x + m2y + n2z = 0.

Solving these equations gives three ratios:

x = y = z

m1n2 - m2n1 n1l2 - n2l1 l1m2 -l2m1

Hence, "the ratios of x, y, z are the ratios of

m1n2 - m2n1, n1l2 - n2l1, l1m2 - l2m1."

Moreover, unless they all vanish, the ratios determine a unique point. That particular point lies on each of the

Reuleaux triangles find practical application in the display of numerical data. Somatocharts are drawn as triangles with curved sides. These charts have three axes radiating from a geometric center.

? = ?(t0) + ( ? ??(t0), -? < ( < ?

The functions ui(t) are also of class C3 and also satisfy the condition that ?u?1? + ?u?2? is not equal to zero for all t in the domain of the curve ?(t).

For the most(prenominal) part, curve width depends on (.

Somatocharts can thus be applied towards the quantification of individual somatotypes. This can be achieved by assign numerical values to each vertex. For instance, the left efflorescence of the somatochart (i.e., the extreme endomorphic vertex) can be represented by the coordinates 7 1 1. Likewise, the top apex (i.e.

, the extreme mesomorphic vertex) whitethorn be assigned the numerical values 171. Lastly, the right apex (i.e., the extreme ectomorphic vertex) receives the numbers 117. The center of the somatochart can be represented by either 4 4 4 or 3 3 3. This value would describe a person with a balanced somatotype.

?d(? = K(s) > 0

The two-dimensional Reuleaux triangle combines some(prenominal) different elements of geometric form. These include the point, the line, intersecting lines, curves, and surfaces. Taken as a whole, the various components of the Reuleaux triangle may be busy as a technique for plotting information. The somatochart, for example, provides a useful implement for analyzing three-numeral data.

Although parametric models may be derived for any number of curves, in that respect are a few which are particularly significant. For example, the representation of a straight line through the end point of a transmitter ( and in the direction of a nonzero vector b is given by the following equation:

as ? ds = ? ds/d( (d() = ? (h + h(**)) d( = ? h d(

Three straight lines having a common point can additionally be described by the equations:

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