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 if no two lines of the regulus of multiple secants of **zg can intersect, then the regulus must be quadratic, or in other words, **zg must be either a Af or a Af curve on a nonsingular quadric surface.

Standard addition can be applied to most analytical techniques and is used instead of a calibration curve to solve the matrix effect problem.

Its light curve can be approximated by adding together a number of sine waves, with periods that range between 0. 8 and 1. 5 hours.

No closed curve can have an asymptote.

Often, low-floor trams are fitted with nonpivoting bogies and many tramway enthusiasts see this as a retrograde step, as it leads to more wear of both track and wheels and also significantly reduces the speed at which a tram can round a curve.

Bézier curves are also used in the time domain, particularly in animation and interface design, e. g., a Bézier curve can be used to specify the velocity over time of an object such as an icon moving from A to B, rather than simply moving at a fixed number of pixels per step.

As the curve is completely contained in the convex hull of its control points, the points can be graphically displayed and used to manipulate the curve intuitively.

Affine transformations such as translation, and rotation can be applied on the curve by applying the respective transform on the control points of the curve.

Another approach, used by modern hardware graphics adapters with accelerated geometry, can convert exactly all Bézier and conic curves ( or surfaces ) into NURBS, that can be rendered incrementally without first splitting the curve recursively to reach the necessary flatness condition.

Writing B < sub > P < sub > i </ sub >, P < sub > j </ sub >, P < sub > k </ sub ></ sub >( t ) for the quadratic Bézier curve defined by points P < sub > i </ sub >, P < sub > j </ sub >, and P < sub > k </ sub >, the cubic Bézier curve can be defined as a linear combination of two quadratic Bézier curves:

* A curve can be split at any point into two subcurves, or into arbitrarily many subcurves, each of which is also a Bézier curve.

* Some curves that seem simple, such as the circle, cannot be described exactly by a Bézier or piecewise Bézier curve ; though a four-piece cubic Bézier curve can approximate a circle ( see Bézier spline ), with a maximum radial error of less than one part in a thousand, when each inner control point ( or offline point ) is the distance horizontally or vertically from an outer control point on a unit circle.

More generally, an n-piece cubic Bézier curve can approximate a circle, when each inner control point is the distance from an outer control point on a unit circle, where t is 360 / n degrees, and n > 2.

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.

An example of this process can be found at Key Length which uses multiple reports to suggest that a symmetric cipher with 128 bits, an asymmetric cipher with 3072 bit keys, and an elliptic curve cipher with 512 bits, all have similar difficulty at present.

A function from the set of real numbers to the real numbers can be represented by a graph in the Cartesian plane ; the function is continuous if, roughly speaking, the graph is a single unbroken curve with no " holes " or " jumps ".

Thus, a child's Skeletal Age `` dots '' may be classified as `` advanced '' when they appear above the middle curve, `` moderate '' when they appear immediately above or below the middle curve, and `` delayed '' when they appear below the lower curve.

For any such square the middle corner of these will be called the vertex of the square and the corner not on the curve will be called the diagonal point of the square.

Let Q be a nonsingular quadric surface bearing reguli Af and Af, and let **zg be a Af curve of order K on Q.

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.

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

By varying the formula, this curve may be moved forward or backward along the coordinates to produce any desired compression strength / density ratio.

The curve on the card the computer spat back at him couldn't be argued with, however.

Not only did this mean that Ford enjoyed much larger demand, but the resulting larger demand also allowed further economies of scale to be exploited, further depressing unit price, which tapped yet another portion of the demand curve.

In three dimensions, a single equation usually gives a surface, and a curve must be specified as the intersection of two surfaces ( see below ), or as a system of parametric equations.

Being behind the technological curve was not entirely surprising ; the chipset used in the 400 and 800 was designed because they were aware the 2600 would be obsolete by the 1980 time frame.

When animators or interface designers talk about the " physics " or " feel " of an operation, they may be referring to the particular Bézier curve used to control the velocity over time of the move in question.

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.

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.

* Conic section, any curve obtained by cutting a conical surface by an arbitrary plane

The market supply curve is obtained by summing the quantities supplied by all suppliers at each potential price.

The market demand curve is obtained by summing the quantities demanded by all consumers at each potential price.

A rough approximation to thermal inertia is sometimes obtained from the amplitude of the diurnal temperature curve ( i. e., maximum minus minimum surface temperature ).

All curves with the same tangent vector will have the same normal curvature, which is the same as the curvature of the curve obtained by intersecting the surface with the plane containing T and u.

The image of simple plane curve can have Hausdorff dimension bigger than one ( see Koch snowflake ) and even positive Lebesgue measure ( the last example can be obtained by small variation of the Peano curve construction ).

For an exact solution, the value of μ at the B value used must be obtained from the core material hysteresis curve.

In growing individuals, serial radiographs are obtained at three-to 12-month intervals to follow curve progression, and, in some instances, MRI investigation is warranted to look at the spinal cord.

If the turbines used have high peak efficiencies, but behave poorly at partial load, less annual performance is obtained than with turbines that have a flat efficiency curve.

The theorem states that any elliptic curve over Q can be obtained via a rational map with integer coefficients from the classical modular curve

The curve obtained by finding the corresponding cusp form, and then constructing a curve from it, is isogenous to the original curve ( but not, in general, isomorphic to it ).

Also, for fast focal ratios, the curve obtained is not sufficiently exact and requires additional hand correction.

That is, the indifference curve tangent to the budget constraint represents the maximum utility obtained utilizing the entire budget of the consumer.

The isomorphisms between fibers obtained in this way will in general depend on the choice of the curve: if they do not then parallel transport along every curve can be used to define parallel sections of E over all of M. This is only possible if the curvature of ∇ is zero.

Given a covariant derivative ∇, the parallel transport along a curve γ is obtained by integrating the condition.

The value of the EER can be easily obtained from the ROC curve.