This approach also allows preserving the curve definition under all linear or perspective 2D and 3D transforms and projections.

A recursive definition for the Bézier curve of degree n expresses it as a point-to-point linear combination of a pair of corresponding points in two Bézier curves of degree n − 1.

One of the next milestones came in 1904, when Helge von Koch, extending ideas of Poincaré and dissatisfied with Weierstrass's abstract and analytic definition, gave a more geometric definition including hand drawn images of a similar function, which is now called the Koch curve ( see Figure 2 ).

If we let Δx and Δy be the distances ( along the x and y axes, respectively ) between two points on a curve, then the slope given by the above definition,

Among other uses, Advanced Covering can be used for end-capping, post-translation corrections, and surface definition from curve data.

The first definition is for a planar curve that extends primarily in length and width, but not in height.

When X is an algebraic curve with field of definition the complex numbers, and if X has no singular points, then these definitions agree and coincide with the topological definition applied to the Riemann surface of X ( its manifold of complex points ).

The definition of elliptic curve from algebraic geometry is connected non-singular projective curve of genus 1 with a given rational point on it.

Indeed, this is essentially an algebraic translation of these geometric terms: the coordinate ring of any affine variety is, by definition, a finitely generated k-algebra, so Noetherian ; moreover curve means dimension one and nonsingular implies ( and, in dimension one, is equivalent to ) normal, which by definition means integrally closed.

Below an equivalent definition will be explained: A world line is a time-like curve in spacetime.

for some integer N ; this is a curve with integer coefficients with an explicit definition.

From the definition of the cubic Bézier curve, we have:

This is the approach taken in the definition of point-price elasticity, which uses differential calculus to calculate the elasticity for an infinitesimal change in price and quantity at any given point on the demand curve:

The Levi-Civita notions of intrinsic derivative and parallel displacement of a vector along a curve make sense on an abstract Riemannian manifold, even though the original motivation relied on a specific embedding, since the definition of the Christoffel symbols make sense in any Riemannian manifold.

For this definition to be correct, we must use an algebraically closed field and a curve C in projective space ( i. e., complete in the sense of algebraic geometry ).

Web colors have an unambiguous colorimetric definition, sRGB, which relates the chromaticities of a particular phosphor set, a given transfer curve, adaptive whitepoint, and viewing conditions.

The definition of the continuity ' Cn requires that the n < sup > th </ sup > derivative of the curve / surface () are equal at a joint.

This is as the definition finds the tangent to the total cost curve at the point q which assumes that costs increase at the same rate as they were at q.

To eliminate the inherent vagueness of this notion Jordan in 1887 introduced the following rigorous definition, which has since been adopted as the precise description of the notion of a continuous curve:

The Content Model is a structured framework that captures the knowledge that underpins the definition of an ICD entity.

This definition captures the notion of self-similarity implied in the name " scale-free ".

The following definition outlines the essential difference between the two: A learning organization actively creates, captures, transfers, and mobilizes knowledge to enable it to adapt to a changing environment.

This captures the main geometric idea, but the full definition is somewhat more complex.

Examples come from fractional ideals in algebraic number theory, so that the definition captures that theory.

This captures Cauchy's definition of continuity.

Norman's definition of ( perceived ) affordances captures the likelihood that the actor will sit on the recliner and throw the softball.

British power system planners and operational staff therefore invariably speak of the Supergrid in this context ; in practice the definition used captures all of the infrastructure owned by the National Grid company in England and Wales, and ( in England and Wales ) no other equipment.

Despite its intuitive notion there is not yet a definition or characterization of centrality indices which captures all of them.

The EPA's definition of the term " captures key elements of the transformations " observed in the animal agriculture sector over the course of the 20th century: " a production process that concentrates large numbers of animals in relatively small and confined places, and that substitutes structures and equipment ( for feeding, temperature controls, and manure management ) for land and labor.

The thesis that the definition of Martin-Löf randomness " correctly " captures the intuitive notion of randomness has been called the " Martin-Löf-Chaitin Thesis "; it is somewhat similar to the Church – Turing thesis.

At this point a working definition of idea is in order, although our first definition will have to be qualified somewhat as we proceed.

In addition, his definition of a tragedy invites our attention, because a serious and important action may very well be one that tests the moral fiber of the author or of the characters.

Bliss found especially useful their " triangle of reference ": the physical thing or " referent " that we perceive would be represented at the right angle ; the meaning that we know by experience ( our implicit definition of the thing ), at the top angle ; and the physical word that we speak or write, at the left angle.

Pope Paul VI spoke of it as " a profession of faith, ... a creed which, without being strictly speaking a dogmatic definition, repeats in substance, with some developments called for by the spiritual condition of our time, the creed of Nicea, the creed of the immortal tradition of the holy Church of God "

In a debate at Orange Coast College, he voiced his support for Proposition 8, which defines marriage in California as only between a man and a woman, and said that he " would suggest not changing the definition of marriage in our society to make a small number of people feel more comfortable.

" R. M. Hare also criticised ethical naturalism because of its fallacious definition of the terms ' good ' or ' right ' explaining how value-terms being part of our prescriptive moral language are not reducible to descriptive terms: " Value-terms have a special function in language, that of commending ; and so they plainly cannot be defined in terms of other words which themselves do not perform this function "

However, the method of cooking these sheets of dough does not correspond to our modern definition of either a fresh or dry pasta product, which only had similar basic ingredients and perhaps the shape.

Another definition of research is given by Creswell who states-" Research is a process of steps used to collect and analyze information to increase our understanding of a topic or issue ".

Now, according to our definition, Tragedy is an imitation of an action that is complete, and whole, and of a certain magnitude … As therefore, in the other imitative arts, the imitation is one when the object imitated is one, so the plot, being an imitation of an action, must imitate one action and that a whole, the structural union of the parts being such that, if any one of them is displaced or removed, the whole will be disjointed and disturbed.

: Two political Sects have arisen within the U. S. the one believing that the executive is the branch of our government which the most needs support ; the other that like the analogous branch in the English Government, it is already too strong for the republican parts of the Constitution ; and therefore in equivocal cases they incline to the legislative powers: the former of these are called federalists, sometimes aristocrats or monocrats, and sometimes tories, after the corresponding sect in the English Government of exactly the same definition: the latter are stiled republicans, whigs, jacobins, anarchists, disorganizers, etc.

Condorcet believed that there was no definition of the perfect human existence and thus believed that the progression of the human race would inevitably continue throughout the course of our existence.

" If Kahnweiler considers Cubism as Picasso and Braque ," writes Daniel Robbins, " our only fault is in subjecting other Cubists ' works to the rigors of that limited definition.

" Hume also took his definition of reason to unorthodox extremes by arguing, unlike his predecessors, that human reason is not qualitatively different from either simply conceiving individual ideas, or from judgments associating two ideas, and that " reason is nothing but a wonderful and unintelligible instinct in our souls, which carries us along a certain train of ideas, and endows them with particular qualities, according to their particular situations and relations.

We must not exclude potato-growing or banking from our definition.

A preliminary observation, in parallel with our definition

Thus a definition reflects the ultimate object of our understanding, and is the foundation of all valid inference.

Because of this, our definition of what music technology encompasses continually expands.

Thus by definition our formula is not a tautology.

We need to give a clear definition to our language ... We know that ebony means black and that phonics refers to speech sounds or the science of sounds.

Even light itself does not have a " velocity " of c in this sense ; the total velocity of any object can be expressed as the sum where is the recession velocity due to the expansion of the universe ( the velocity given by Hubble's law ) and is the " peculiar velocity " measured by local observers ( with and, the dots indicating a first derivative ), so for light is equal to c (- c if the light is emitted towards our position at the origin and + c if emitted away from us ) but the total velocity is generally different than c .( Davis and Lineweaver 2003, p. 19 ) Even in special relativity the coordinate speed of light is only guaranteed to be c in an inertial frame, in a non-inertial frame the coordinate speed may be different than c ; in general relativity no coordinate system on a large region of curved spacetime is " inertial ", but in the local neighborhood of any point in curved spacetime we can define a " local inertial frame " and the local speed of light will be c in this frame, with massive objects such as stars and galaxies always having a local speed smaller than c. The cosmological definitions used to define the velocities of distant objects are coordinate-dependent-there is no general coordinate-independent definition of velocity between distant objects in general relativity ( Baez and Bunn, 2006 ).

Otherwise our definition is strictly weaker, being permissive enough to make all nonzero R-submodules of R — i. e., integral ideals — fractional ideals.

The Protestant reformer John Calvin said that repentance " may be justly defined to be “ a true conversion of our life to God, proceeding from a serious fear of God, and consisting in the mortification of the flesh and of the old man, and in the vivification of the Spirit .” He further said that " it will be useful to amplify and explain the definition we have given ; in which there are three points to be particularly considered.

Cultural theorist Kwame Anthony Appiah criticised DuBois for this definition of racialism in ' My Father's House ' ( 1992 ) where he defines racialism as "... the view … that there are heritable characteristics, possessed by members of our species, which allow us to divide them into a small set of races, in such a way that all the members of these races share certain traits and tendencies with each other that they do not share with members of any other race.