If one assumes that the average flux did not change between measurements, a mass-distribution curve is obtained which relates the flux of particles larger than a given radius to the inverse 7/2 power of the radius.

Consider a simple, closed, plane curve C which is a real-analytic image of the unit circle, and which is given by Af.

In the following paper it is shown that in a certain definite sense, exactly an odd number of squares can be inscribed in every such curve which does not contain an infinite number of inscribed squares.

If the vertex is at Af, and if the interior of C is on the left as one moves in the direction of increasing t, then every such corner can be found from the curve obtained by rotating C clockwise through 90-degrees about the vertex.

But Af is just the curve Af translated without rotation through a small arc, for Af is always obtained by rotating C through exactly 90-degrees.

The arc is itself a segment of an analytic curve.

Then in 2 we show that any line involution with the properties that ( A ) It has no complex of invariant lines, and ( B ) Its singular lines form a complex consisting exclusively of the lines which meet a twisted curve, is necessarily of the type discussed in 1.

The order of this congruence is Af, since Af secants of a curve of symbol ( B ) on a quadric surface pass through an arbitrary point.

For the lines of any plane, **yp, meeting Q in a conic C, are transformed into the congruence of secants of the curve C' into which C is transformed in the point involution on Q.

Each generator, **yl, of Af is also exceptional, for each is transformed into the entire congruence of secants of the curve into which that generator is transformed by the point involution on Q.

This curve is of symbol Af since it meets **yl, and hence every line of Af in the Af invariant points on **yl and since it obviously meets every line of Af in a single point.

Hence C' is a Af curve on Q.

By ( 1 ), the image of this pencil is a ruled surface of order Af which is met by the plane of the pencil in a curve, C, of order Af.

To avoid this contradiction it is necessary that C be composite, with the secant of **zg and a curve of order Af as components.

We now observe that the case in which **zg is a Af curve on a quadric is impossible if the complex of singular lines consists exclusively of the lines which meet Aj.

This contradicts the preceding observations, and so, under the assumption of this paper, we must reject the possibility that **zg is a Af curve on a quadric surface.

Continuing with the case in which **zg is a Af curve on a quadric Q, we first observe that the second regulus of Q consists precisely of the lines which join the two free intersections of **zg and the planes through any one of the multiple secants.

We assume that average total unit cost in the relevant region of operation is constant with respect to quantity produced ( the average cost curve is horizontal, and therefore is identical with the marginal cost curve ), and is the same for every firm ( and therefore for the industry ).

Given points P < sub > 0 </ sub > and P < sub > 1 </ sub >, a linear Bézier curve is simply a straight line between those two points.

In Geometric measure theory such a smooth curve as the circle that can be approximated by small straight segments with a definite limit is termed a rectifiable curve.

With hinged, fixed or free end support conditions the deflected shape in neutral equilibrium of an initially straight column with uniform cross section throughout its length always follows a partial or composite sinusoidal curve shape, and the critical load is given by

When the rode is slack, the catenary curve presents a lower angle of pull on the anchor or mooring device than would be the case if it were nearly straight.

When a parabola is rolled along a straight line, the roulette curve traced by its focus is a catenary.

* Cable Sag Error Calculator – Calculates the deviation from a straight line of a catenary curve and provides derivation of the calculator and references.

For example, if u < sub > 1 </ sub > is an eigenvector of A, with a real eigenvalue smaller than one, then the straight lines given by the points along α u < sub > 1 </ sub >, with α ∈ R, is an invariant curve of the map.

To elaborate, in trying to find the length of a wavy non-fractal curve, one could find straight segments of some measuring tool small enough to lay end to end over the waves, where the pieces could get small enough to be considered to conform to the curve in the normal manner of measuring with a tape measure.

But in measuring a wavy fractal curve such as the one in Figure 2, one would never find a small enough straight segment to conform to the curve, because the wavy pattern would always re-appear, albeit at a smaller size, essentially pulling a little more of the tape measure into the total length measured each time one attempted to fit it tighter and tighter to the curve.

Some media have an index of refraction which varies gradually with position and, thus, light rays curve through the medium rather than travel in straight lines.

On a plot of torque versus runner speed, the torque curve is straight between these two points, ( 0, pQDV < sub > i </ sub >) and ( V < sub > i </ sub >, 0 ).

Radius is a straight line or distance from the center to the edge of a curve.

All the samples show loss of lead isotopes, but the intercept of the errorchron ( straight line through the sample points ) and the concordia ( curve ) shows the correct age of the rock.

This can be seen in the concordia diagram, where the samples plot along an errorchron ( straight line ) which intersects the concordia curve at the age of the sample.

For instance, Tarnow shows that the recall probability vs. latency curve is a straight line from 6 to 600 seconds ( ten minutes ), with the probability of failure to recall only saturating after 600 seconds.

In geometry, the tangent line ( or simply the tangent ) to a plane curve at a given point is the straight line that " just touches " the curve at that point — that is, coincides with the curve at that point and, near that point, is closer to the curve that any other line passing through that point.

More precisely, a straight line is said to be a tangent of a curve

These are defined by a simple involutorial transformation of the points in which a general line meets a nonsingular quadric surface bearing a curve of symbol Af.

A conic, C, being a ( 1, 1 ) curve on Q, meets the image of any line of Af, which we have already found to be a Af curve on Q, in Af points.

The preceding observations make it clear that there exist line involutions of all orders greater than 1 with no complex of invariant lines and with a complex of singular lines consisting exclusively of the lines which meet a twisted curve Aj.

The single curve line represents a specific formulation in a test example.

In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity.

Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors.

In some contexts, such as algebraic geometry, an asymptote is defined as a line which is tangent to a curve at infinity.

" The term was introduced by Apollonius of Perga in his work on conic sections, but in contrast to its modern meaning, he used it to mean any line that does not intersect the given curve.

The idea that a curve may come arbitrarily close to a line without actually becoming the same may seem counter to everyday experience.

The representations of a line and a curve as marks on a piece of paper or as pixels on a computer screen have a positive width.

But these are physical representations of the corresponding mathematical entities ; the line and the curve are idealized concepts whose width is 0 ( see Line ).

Oblique asymptotes are diagonal lines so that the difference between the curve and the line approaches 0 as x tends to +∞ or −∞.

The simplest method for scan converting ( rasterizing ) a Bézier curve is to evaluate it at many closely spaced points and scan convert the approximating sequence of line segments.

A common adaptive method is recursive subdivision, in which a curve's control points are checked to see if the curve approximates a line segment to within a small tolerance.

The corresponding form of the fundamental theorem of calculus is Stokes ' theorem, which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve.

* Chord ( geometry ), a line segment joining two points on a curve