Image Watt curve animated.gif right In mathematics, Watt s curve is a circular algebraic curve tricircular algebraic curveplane algebraic curve of sextic degree six . It is generated by two circles of radius ... curve as the circles rotate. It arose in connection with James Watt s pioneering work on the steam engine. The equation of the curve can be given in polar coordinates as math r 2 b 2 left a sin theta pm sqrt c 2 a 2 cos 2 theta right 2. math Derivation Polar coordinates The polar equation for the curve can be derived as follows ref See Catalan and Rutter ref Working in the complex plane , let the centers .... , math Form of the curve The construction requires a quadrilateral with sides 2 a , b , 2 c , b . Any side must be less than the sum of the remaining sides, so the curve is empty at least in the real plane unless a b c and c b a . The has a crossing point at the origin if there is a triangle with sides a , b and c . Given the previous conditions, this means that the curve crosses the origin if and only if b a c . If b a c then two branches of the curve meet at the origin with a common vertical tangent, making it a quadruple point. Given b a c , the shape of the curve is determined by the relative magnitude of b and d . If d is imaginary, that is if a sup 2 sup b sup 2 sup c sup 2 sup then the curve has fthe form of a figure eight. If d is 0 then the curve is a figure eight with two branches of the curve having a common horizontal tangent at the origin. If 0 d b then the curve has two additional double points at d and the curve crosses itself at these points. The overall shape of the curve is pretzel like in this case. If d b then a c and the curve decomposes into a circle of radius b and a Hippopede lemniscate of Booth , a figure eight shaped curve. A special case of this is a c ... to the Cartesian equation of the curve, but the curve does not cross these points and they are acnode s. The curve again has a figure eight shape though the shape is distorted if d is close to b ... more details
If X is a plane algebraic curve then the degree of the dual is the number of points intersection with a line in the dual plane. Since a line in the dual plane corresponds to a point in the plane ...Image Dual curve.svg thumb right 300px Curves, dual to each other see below for Properties of dual curve properties . In projective geometry , a dual curve of a given planecurve C is a curve in the Duality projective geometry dual projective plane consisting of the set of lines tangent to C . There is a map from a curve to its dual, sending each point to the point dual to its tangent line. If C is algebraic then so is its dual and the degree of the dual is known as the class of the original curve ... . The construction of the dual curve is the geometrical underpinning for the Legendre transformation ... y , z 0 be the equation of a curve in homogeneous coordinates . Let Xx Yy Zz 0 be the equation ... is tangent to the curve can be expressed in the form F X , Y , Z 0 which is the tangential equation of the curve. Let p , q , r be the point on the curve, then the equation of the tangent ... y p, q, r z frac partial f partial z p, q, r 0. math So Xx Yy Zz 0 is a tangent to the curve if math ... with Xp Yq Zr 0, gives the equation in X , Y and Z of the dual curve. For example, let C be the Conic ... defined curve its dual curve is defined by the following parametric equations math X frac ... where these tangents touch the curve are the points of intersection between the curve and the polar curve with respect to the given point. If the degree of the curve is d then the degree of the polar ... d d 1 . The dual of a line a curve of degree 1 is an exception to this and is taken to be a point ... point. If X is smooth, i.e. there are no Singular point of a curve singular points then the dual of X ... to two points of a conic, as that requires degree 4 , and tangent line varies smoothly as the curve ... an inflection point in the original curve, which requires degree 3 . For curves with singular ... more details
In geometry, a W curve is a curve in projective space projective n space that is invariant mathematics invariant under a 1 parameter group of projective transformations . W curves were first investigated by Felix Klein and Sophus Lie in 1871, who also named them. W curves in the real projective plane can be constructed with straightedge alone. Many well known curves are W curves, among them conic section conics , logarithmic spiral s, powers like y x sup 3 sup , logarithm s and the helix , but not e.g. the sine . W curves occur widely in the realm of plants. Image PlaneWcurve.svg thumb left alt caption 500px A typical plane W curve with source O and sink Y Name The W stands for the German Wurf &ndash a throw &ndash which in this context refers to a series of four points on a line. A 1 dimensional W curve read the motion of a point on a projective line is determined by such a series. The German W Kurve sounds almost exactly like Weg Kurve and the last can be translated by path curve . That is why in the English literature one often finds path curve or pathcurve . References Felix Klein and Sophus Lie Ueber diejenigen ebenen Curven... in Mathematische Annalen, Band 4, 1871 online available at the http gdz.sub.uni goettingen.de dms load img ?PPN PPN235181684 0004 University of Goettingen For an introduction on W curves and how to draw them, see Lawrence Edwards Projective Geometry , Floris Books 2003, ISBN 0 86315 393 3 On the occurrence of W curves in nature see Lawrence Edwards The vortex of life , Floris Books 1993, ISBN 0 86315 148 5 For an algebraic classification of 2 and 3 dimensional W curves see http www.mathart.nl Pathcurves1010.pdf Classification of pathcurves http edwardsgeometry.nl ktmlliterf files uploads Externe 20artikelen ClassPCv5.pdf Classification of pathcurves Georg Scheffers 1903 Besondere transzendente Kurven , Klein s encyclopedia Band 3&ndash 3. Category Curves Category Projective geometry de W Kurve nl W kromme ... more details
this process indefinitely produces a tessellation of the plane. The curve itself can likewise be extended to an infinite curve filling the whole plane. align center Image Gosper Island Tesselation ...The Gosper curve , named after Bill Gosper , also known as the flowsnake a spoonerism of snowflake , is a space filling curve . It is a fractal object similar in its construction to the dragon curve and the Hilbert curve . Image Gosper curve 3.svg Image Gosper curve 1.svg A fourth stage Gosper curve The line from the red to the green point shows a single step of the Gosper curve construction. A Logo programming language Logo program to draw the Gosper curve using turtle graphics http logo.twentygototen.org mJjiNzK0 online version code to rg st ln make st st 1 make ln ln 2.6457 if st 0 rg st ln rt 60 gl st ln rt 120 gl st ln lt 60 rg st ln lt 120 rg st ln rg st ln lt 60 gl st ln rt 60 if st 0 fd ln rt 60 fd ln rt 120 fd ln lt 60 fd ln lt 120 fd ln fd ln lt 60 fd ln rt 60 end to gl st ln make st st 1 make ln ln 2.6457 if st 0 lt 60 rg st ln rt 60 gl st ln gl st ln rt 120 gl st ln rt 60 rg st ln lt 120 rg st ln lt 60 gl st ln if st 0 lt 60 fd ln rt 60 fd ln fd ln rt 120 fd ln rt 60 fd ln lt 120 fd ln lt 60 fd ln end code The program can be invoked, for example, with code rg 4 300 code , or alternatively code gl 4 300 code . The constant 2.6457 in the program code is an approximation of 7. Properties The space filled by the curve is called the Gosper island . The first few iterations of it are shown below align center Image Gosper Island 0.svg 180px Image Gosper Island 1.svg 180px Image Gosper Island 2.svg 180px Image Gosper Island 3.svg 180px Image Gosper Island 4.svg 180px The Gosper Island can tessellation tile the Plane mathematics plane . In fact, seven copies of the Gosper island can be joined together to form a shape that is Similarity geometry similar , but scaled up by a factor of 7 in all dimensions. As can be seen from the diagram below, performing this operation ... more details
In mathematics , in the field of complex geometry , a holomorphic curve in a complex manifold M is a non constant holomorphic map f from the complex plane to M . ref name Shiffman harvtxt Shiffman 1977 , p.553 ref Nevanlinna theory addresses the question of the distribution of values of a holomorphic curve in the complex projective line . ref name Shiffman ref cite book author Min Ru title Nevanlinna Theory and its Relation to Diophantine Approximation publisher World Scientific isbn 981 02 4402 9 year 2001 ref See also Pseudoholomorphic curve Notes reflist References cite journal title Holomorphic curves in algebraic manifolds journal Bulletin of the American Mathematical Society volume 83 number 4 year 1977 author B. Shiffman pages 553 568 url http www.ams.org bull 1977 83 04 S0002 9904 1977 14323 1 S0002 9904 1977 14323 1.pdf ref harv Category Complex manifolds mathanalysis stub ... more details
curves of every curve in the index. Differential transforms of plane curves Category Curves Category ... the red parabola across the dashed circle . In geometry , an inverse curve of a given curve C is the result ... on the ray OQ and OP PQ k sup 2 sup . The Inverse of the curve C is then the locus of P ... transformation, so the inverse of an inverse curve with respect to the same circle is the original curve ... So the inverse of the curve determined by f x , y 0 with respect to the unit circle ... of an algebraic curve of degree n is also algebraic of degree at most 2 n . Similarly, the inverse of the curve defined Parametric equation parametrically by the equations math x x t , y y t math ... y t x t 2 y t 2 . math This implies that the inverse of a rational curve is also rational. More generally, the inverse of the curve determined by f x , y 0 with respect to the circle ... X a 2 Y b 2 right 0. math The inverse of the curve defined parametrically by math x x t , y y t math ... 1 R , theta Theta. math So the inverse of the curve f r , &theta 0 is determined by f 1 R , &Theta 0 and the inverse of the curve r g &theta is r 1 g &theta . Degrees As noted above, the inverse of a curve of degree n has degree at most 2 n . The degree is exactly 2 n unless the original curve passes through the point of inversion or it s Circular algebraic curve circular , meaning that it contains the circular points, 1, i , 0 , when considered as a curve in the complex projective plane. Specifically, if C is p circular of degree n , and if the center of inversion is a singularity of order q on C , then the inverse curve will be an n p q circular curve of degree 2 n 2 p q and the center of inversion is a singularity of order n 2 p on the inverse curve. Here q 0 if the curve does not contain contain the center of inversion and q 1 if the center of inversion ... more details
Butterfly curve may refer to Butterfly curve algebraic , a curve defined by a trinomial Butterfly curve transcendental , a curve based on sine functions disambig Category Mathematical disambiguation es Curva mariposa it Curva a farfalla ... more details
visible at the peaks of the torsion function. Properties A planecurve with non vanishing curvature has zero torsion at all points. Conversely, if the torsion of a regular curve with non vanishing curvature is identically zero, then this curve belongs to a fixed plane. The curvature and the torsion of a helix are constant. Conversely, any space curve with constant non zero curvature and constant ... handed one. Alternative description Let r r t be the parametric equation of a space curve. Assume that this is a regular parametrization and that the curvature of the curve does not vanish. Analytically ... more details
x 3 3xy 2 , math and is therefore a algebraic curveplane algebraic curve of degree four. In polar coordinates this becomes math r 4 18a 2r 2 27a 4 8ar 3 cos 3 theta ,. math The curve has three singularities ...File Deltoid2.gif right thumb 500px The red curve is a deltoid. In geometry , a deltoid , also known as a tricuspoid or Steiner curve , is a hypocycloid of three cusp singularity cusp s. In other words, it is the Roulette curve roulette created by a point on the circumference of a circle as it rolls without slipping along the inside of a circle with three times its radius. It can also be defined as a similar roulette where the radius of the outer circle is frac 3 2 times that of the rolling circle. It is named after the Greek letter Delta letter delta which it resembles. More broadly, a deltoid can refer to any closed figure with three vertices connected by curves that are concave to the exterior, making the interior points a non convex set. http www.btinternet.com se16 js halfarea.htm Equations A deltoid can be represented up to rotation and translation by the following parametric equations math x 2a cos t a cos 2t , math math y 2a sin t a sin 2t , math where a is the radius of the rolling ... that the curve is rational which implies it has geometric genus genus zero. A line segment can slide ... the deltoid twice while each end travels around it once. The dual curve of the deltoid is math x ... by an imaginary rotation y iy, giving the curve math x 3 x 2 3x 1 y 2 0 , math with a double point at the origin of the real plane. Area and perimeter The area of the deltoid is math 2 pi a 2 math ... Steiner who described the shape and symmetry of the curve in 1856. ref Lockwood ref The Envelope ... 108. ref http www.btinternet.com se16 js halfarea.htm See also Astroid , a curve with four cusps .... Dennis Lawrence title A catalog of special plane curves publisher Dover Publications year 1972 isbn ... Math matiques Remarquables in French springer title Steiner curve id S s087650 last Sokolov first ... more details
curve can tile the plane in many ways. center gallery Image Dragon tiling1.svg 1st element ...Image Dragon curve animation.gif right 300 px A dragon curve is any member of a family of Self similarity ... system s. Heighway dragon Image Fractal dragon curve.jpg thumb Heighway dragon curve The Heighway ... right 400px Recursive construction of the curve It can be written as a Lindenmayer system with angle ... angle and with a rotation of 45 alternatively to the right and to the left Image Dragon curve iterations ... of the following iterated function system in the complex plane math f 1 z frac 1 i z 2 math math f ... dragon curve from one end to the other, one encounters a series of 90 degree turns, some to the right ... method of creating models of iterations of the Heighway dragon curve by paper folding folding a strip ... . Image Dragon curve paper strip.png center 800px This pattern also gives a method for determining ... dragon curve has simple dimensions Image Dimensions fractale dragon.gif none 500px Its surface is also ... fm8.html ref The curve never crosses itself. Many self similarities can be seen in the Heighway dragon curve. The most obvious is the repetition of the same pattern tilted by 45 and with a reduction ... curve . The fractal dimension of its boundary has been approximated numerically by Chang & Zhang ... of the Dragon curve ref . In fact it can be found analytically ref http demonstrations.wolfram.com ... curve. ref math log 2 left frac 1 sqrt 3 73 6 sqrt 87 sqrt 3 73 6 sqrt 87 3 right cong 1.523627086202492 ... with 4 curves Image Dragon tiling dragon.svg The dragon curve can tile itself Image Dragon tiling4.svg ... tiling6.svg 3rd element with 2 curves Image Full tiling dragon.svg Example of plane tiling Image Full tiling dragon2.svg Example of plane tiling Image Full tiling dragon3.svg Example of plane tiling ... spiral. 4 of these spirals with rotation 90 tile the plane. gallery center Twindragon The twindragon ... 0,1,1 i math . Image L vy s dragon curve IFS .jpg thumb Twindragon curve. Image Twindragon.png thumb ... more details
Image Kanna2.gif frame A Japanese plane in use A hand plane is a tool for shaping wood . When powered ... surface, or sole , of the plane slices off shavings of wood. A large, flat sole on a plane guides ... or both hands, Japanese plane s are pulled toward the body, not pushed away. Woodworking machinery ... a uniform, smooth surface on all four sides of the lumber and may also plane the edges. History ... Europe and Asia. The earliest known examples of the woodworking plane have been found in Pompeii .... Hampton, E. Clifford Planecraft , page 9. C. and J. Hampton Ltd. 1959 ref A Roman plane iron used for cutting ... were later purchased by Stanley Rule & Level, now Stanley Works . The original Bailey plane Bailey design ... and Bedrock plane Bedrock designs became the basis for most modern metal hand plane designs manufactured .... Parts of a plane Image Plane parts.jpg thumb 300px Parts of a plane Two styles of plane are shown with some parts labeled. The top of the image is a bench plane the bottom is a block plane. Image Bench plane iron.jpg thumb 200px A bench plane iron with chipbreaker. A The mouth is an opening in the bottom of the plane down through which the blade extends, and up through which wood shavings pass. B ... . C The lever cap holds the blade down firmly to the body of the plane. D The depth adjustment ... of the plane. F The chipbreaker or Cap iron serves to make the blade more rigid and to curl and break ... on the rear of the plane. Some aficionados object to the use of the word tote preferring handle . I ... this. J The frog is a sliding iron wedge that holds the plane iron at the proper angle. It slides ... when the plane iron is removed. Some planes, such as the Stanley Black & Decker Stanley Bedrock plane Bedrock line and the bench planes made by Lie Nielsen Toolworks Lie Nielsen have a screw mechanism ... thumb 200px Modern wooden plane Image Strug gladzik.jpg thumb right 200px A smoothing plane Image Stanley transitional jointer plane.jpg thumb right 200px Stanley No. 32 transitional jointer plane ... more details
wiktionarypar curve A curve is a geometrical object in mathematics. Curve may also refer to Music Curve band , an English music group Curve Doc Walker album Curve Doc Walker album , a 2001 album by Doc Walker Curve Our Lady Peace album Curve Our Lady Peace album , a 2012 album by Our Lady Peace Others Curve magazine Curve magazine , a U.S. lesbian magazine Curve design magazine Curve design magazine , an industrial design magazine Curve theatre , a theatre in Leicester, United Kingdom Curve tonality , a software technique for image manipulation Curve, fragrance by Liz Claiborne Grading curve , a system of grading students BlackBerry Curve , a series of phones from Research In Motion French curve , a template made out of plastic, metal or wood used to draw smooth curves Flat spline , a very flexible rule used to draw curves Curves may refer to Curves International , an international fitness franchise Female body shape See also The Curve disambiguation Curvature Curveball disambig de Curve eu Kurba argipena fr Curve ko ja ... more details
polar of an algebraic planecurve C of degree n with respect to a point Q is an algebraic curve ...About a concept in algebraic geometry concept in aviation that goes by that name Polar curve aviation curves given in polar coordinates Polar coordinate system Polar equation of a curve Image PolarEllipticCurvePlot.svg thumb 300px right The elliptic curve E 4 Y sup 2 sup Z X sup 3 sup &minus XZ sup 2 sup in blue, and its polar curve E 4 Y sup 2 sup 2.7 X sup 2 sup &minus 2 XZ &minus 0.9Z sup 2 sup for the point Q 0.9, 0 in red. The black lines show the tangents to E at the intersection points of E ... the relationship between the curve and its Dual curve dual , for example in the derivation ... of degree n 1 and sub Q sub f x, y, z 0 defines a curve of degree n 1 called ... variety non singular point on the curve C then the equation of the tangent at P is math x partial ... of f are all 0 so the first polar contains these points as well. Class of a curve The class of C may ... an upper bound of n n 1 on the class of a curve of degree n . The class may be computed ... z 0. This is a curve of degree n p . When p is n 1 the p th polar is a line called the polar line of C with respect to Q . Similarly, when p is n 2 the curve is called the polar conic ... and Q on L . The locus of points whose polar lines pass through P is the first polar of P and this is a curve ... polar of Q and this is also a curve of degree n 1 . The polar line of a point is L if and only if it contains ... 2 f partial z 2 end bmatrix , math vanishes. Therefore the equation H f 0 defines a curve, the locus of points whose polar conics are degenerate, of degree 3 n 2 called the Hessian curve of C . See ... 16ff. url http books.google.com books?id yUxtAAAAMAAJ cite book title Higher Plane Curves first ... Ivanov first A.B. springer title Hessian algebraic curve id H h047150 last Ivanov first A.B. Algebraic ... more details
Sierpi ski curves are a recursively defined sequence of Geometric continuity continuous closed plane fractal curve s discovered by Wac aw Sierpi ski , which in the limit math n rightarrow infty math completely fill the unit square thus their limit curve, also called the Sierpi ski curve , is an example of a space filling curve . Because the Sierpi ski curve is space filling, its Hausdorff dimension in the limit math n rightarrow infty math is math 2 math . br The euclidean distance Euclidean length of math S n math is math l n 2 over 3 1 sqrt 2 2 n 1 over 3 2 sqrt 2 1 over 2 n math , i.e., it grows ... . border 0 valign top Image Sierpinski Curve 1.png thumb 200px Sierpi ski curve of first order valign top Image Sierpinski Curve 2.png thumb 200px Sierpi ski curves of orders 1 and 2 valign top Image Sierpinski Curve 3.png thumb 200px Sierpi ski curves of orders 1 to 3 Uses of the curve The Sierpi ski curve is useful in several practical applications because it is more symmetrical than other commonly ... on the Sierpi ski curve ref Platzman89 . To do this requires two steps First compute an inverse ... curve is a continuous map of the unit interval onto a unit square and so a pseudo inverse maps the unit ... structure of the curve. Here is a function coded in Java that will compute the relative position of any point on the Sierpi ski curve that is, a pseudo inverse value . It takes as input the coordinates ..., maxLevel, 2 code 1, x, y return code source Drawing the curve The following Java programming language Java applet draws a Sierpi ski curve by means of four methods that recursion recursively ... Logo programming language Logo program draws a Sierpi ski curve by means of recursion . code ... See also Commons Sierpi ski curve Hilbert curve Koch snowflake Moore curve Peano curve Sierpi ski arrowhead curve List of fractals by Hausdorff dimension Recursion computer science Recursion computer science Sierpinski triangle Category Fractal curves Sierpinski curve Category Science and technology ... more details
In mathematics , a positively oriented curve is a planar simple closed curve that is, a curve in the plane whose starting point is also the end point and which has no other self intersections such that when traveling on it one always has the curve interior to the left and consequently, the curve exterior to the right . If in the above definition one interchanges left and right, one obtains a negatively oriented curve . Crucial to this definition is the fact that every simple closed curve admits a well defined interior that follows from the Jordan curve theorem . All simple closed curves can be classified as negatively oriented clockwise , positively oriented counterclockwise , or orientability non orientable . The Inner Outer labeling inner loop of a beltway road in the United States or other countries where people drive on the right side of the road would be an example of a negatively oriented clockwise curve. A circle oriented counterclockwise is an example of a positively oriented curve. The same circle oriented clockwise would be a negatively oriented curve. The concept of orientation of a curve is just a particular case of the notion of orientation mathematics orientation of a manifold that is, besides orientation of a curve one may also speak of orientation of a surface , hypersurface , etc. . Here, the interior and the exterior of a curve both inherit the usual orientation of the plane. The positive orientation on the curve is then the orientation it inherits as the boundary of its interior the negative orientation is inherited from the exterior. Orientation of a simple polygon Image determining orientation.png right Selecting reference points. In two dimensions, given ... polygon complex polygon or for any self intersecting curve there is no natural notion of the interior ... http mathworld.wolfram.com CurveOrientation.html Curve orientation at MathWorld DEFAULTSORT Curve Orientation Category Curves Category Orientation Category Polygons ... more details
element. Often the curve itself, without O specified, is called an elliptic curve. Any elliptic curve can be written as a plane algebraic curve defined by an equation of the form math y ..., the above equation is not quite general enough to comprise all non singular cubic planecurve cubic ... 3 sup x and y sup 2 sup x sup 3 sup x 1 In this context, an elliptic curve is a planecurve defined ... 3,3 sup 2 sup For a 0 and b 0 it s not smooth and therefore not an elliptic curve. In mathematics , an elliptic curve is a differentiable manifold smooth , algebraic variety Projective varieties projective algebraic curve of Genus of an algebraic curve genus one, on which there is a specified point O . An elliptic curve is in fact an abelian variety that is, it has a multiplication defined algebraically ... O is actually the point at infinity in the projective plane . If y sup 2 sup P x , where P is any polynomial of degree three in x with no repeated roots, then we obtain a nonsingular planecurve of genus mathematics genus one, which is thus also an elliptic curve. If P has degree four and is squarefree this equation again describes a planecurve of genus one however, it has no natural choice of identity element. More generally, any algebraic curve of genus one, for example from the intersection ... curve, provided that it has at least one rational point . Using the theory of elliptic function s, it can ... into the complex projective plane . The torus is also an abelian group, and in fact this correspondence ... find applications in cryptography see the article elliptic curve cryptography and integer factorization . An elliptic curve is not an ellipse see elliptic integral for the origin of the term. Topologically, an elliptic curve is a torus . Elliptic curves over the real numbers Although the formal definition of an elliptic curve is fairly technical and requires some background in algebraic geometry ... is called a Weierstrass equation . The definition of elliptic curve also requires that the curve ... more details
, P sub 2 sub and P sub 3 sub in the plane or in higher dimensional space define a cubic B zier curve ...Image Bezier curve.svg thumb right 300px Cubic B zier curve A B zier curve is a parametric curve frequently ... and interface design, e.g., a B zier curve can be used to specify the velocity over time of an object ... of an operation, they may be referring to the particular B zier curve used to control the velocity ... Image B zier curve in Adobe Illustrator CS2.png left thumb B zier path in Adobe Illustrator B zier curves are widely used in computer graphics to model smooth curves. As the curve is completely contained ... the curve intuitively. Affine transformation s such as translation geometry translation , and rotation can be applied on the curve by applying the respective transform on the control points of the curve. Quadratic and cubic B zier curves are most common higher degree curves are more computationally ... method for scan converting Rasterisation rasterizing a B zier curve is to evaluate it at many ... too far apart. Conversely it may generate too many points in areas where the curve is close to linear. 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. If not, the curve ... the curve is flat enough to be drawn as a series of linear or circular segments. The exact splitting ... precision and to avoid non monotonic local changes of curvature. The smooth curve feature of charts ... the curve recursively to reach the necessary flatness condition. This approach also allows preserving the curve definition under all linear or perspective 2D and 3D transforms and projections ... ref Examination of cases A B zier curve is defined by a set of control points P sub 0 sub through ... control points are always the end points of the curve however, the intermediate control points if any generally do not lie on the curve. Linear B zier curves Given points P sub 0 sub and P sub 1 sub ... more details
&ndash we also ensure that the contour integral of over any closed curve lying in the cut plane ... plane. The distance along the light blue line from the origin to the point z is the modulus or absolute value of z . The angle &phi is the argument of z . In mathematics , the complex plane or z plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis . It can be thought of as a modified Cartesian plane , with the real part ... complex plane , it is not the only one possible. Alternatives include the split complex number split complex plane and the dual number s, as introduced by Quotient ring Alternative complex planes quotient rings . ref The concept of the complex plane allows a geometric interpretation of complex numbers ... 1 acts as a rotation. The complex plane is sometimes called the Argand plane because it is used in Argand ... plane. Notational conventions In complex analysis the complex numbers are customarily represented by the symbol ... Cartesian plane . In the Cartesian plane the point x , y can also be represented in polar coordinates ... right . , math In the Cartesian plane it may be assumed that the inverse trigonometric function arctangent ... form here . ref In the complex plane these polar coordinates take the form math z x iy z left ..., because the exponential function On the complex plane complex exponential function is periodic, with period ... curve is important &ndash reversing the direction in which the curve is traversed multiplies the value ... that is, with functions that map some subset of the complex plane into some other possibly overlapping, or even identical subset of the complex plane. Here it is customary to speak of the domain mathematics domain of f z as lying in the z plane, while referring to the range mathematics range or image of f z as a set of points in the w plane. In symbols we write math z x iy qquad f z w u iv , math and often think of the function f as a transformation of the z plane with coordinates x , y into the w ... more details
Image Hilbert curve.gif right thumb 261px First 8 steps toward building the Hilbert curve A Hilbert curve also known as a Hilbert space filling curve is a Geometric continuity continuous fractal space filling curve first described by the German mathematician David Hilbert in 1891, ref D. Hilbert http ... dms img ?PPN PPN235181684 0036&DMDID dmdlog13 Sur une courbe, qui remplit toute une aire plane ... 2 . math H n math is the math n math th approximation to the limiting curve. The euclidean distance .... Images gallery Image Hilbert curve 1.svg Hilbert curve, first order Image Hilbert curve 2.svg Hilbert curves, first and second orders Image Hilbert curve 3.svg Hilbert curves, first to third orders gallery gallery Image Hilbert512.gif Hilbert curve in three dimensions Image Hilbert3d step3.png 3 D Hilbert curve with color showing progression gallery Applications and mapping algorithms Both the true Hilbert curve and its discrete approximations are useful because they give a mapping between ... square, and d is the distance along the curve when it reaches that point, then points that have ... when mapping from a 2D space to a 1D space. But the Hilbert curve does a good job of keeping ..., the Hilbert curve is widely used in computer science. For example, the range of IP address IP addresses used by computers can be mapped into a picture using the Hilbert curve. Code to generate the image would map from 2D to 1D to find the color of each pixel, and the Hilbert curve is sometimes used ... each pixel added to the next pixel along the Hilbert curve. Code to do this would map from 1D to 2D, and the Hilbert curve is sometimes used because it does not create the distracting patterns that would ... to be used instead of z order curve Z order because it has better locality preserving behavior .... Representation as Lindenmayer system The Hilbert Curve can be expressed by a rewriting rewrite system ... . Arthur Butz ref A.R. Butz Alternative algorithm for Hilbert s space filling curve. IEEE Trans. On Computers ... more details
wiktionary bell shaped bell curve bell curves Bell curve may refer to A Gaussian function , a specific kind of function whose graph is a bell shaped curve Normal distribution Normal Gaussian distribution , whose density function is a Gaussian function The Bell Curve , a 1994 book by Richard J. Herrnstein and Charles Murray Bell curve grading , a method of evaluating scholastic performance disambiguation DEFAULTSORT Bell Curve Category Mathematical disambiguation ... more details
Power curve may refer to The power band of an internal combustion engine, the range of speeds in which it operates efficiently The Drag physics Drag in aerodynamics power curve in aerodynamics, a characteristic curve of drag vs. airspeed for airfoils disambig ... more details
In geometry an imaginary curve is an algebraic curve which contains a finite number of imaginary point s. See also Imaginary point Real point Imaginary line mathematics Real line Real curve geometry stub Category Projective geometry ... more details
In geometry a real curve is an algebraic curve that contains an infinite number of real point s. See also Imaginary point Real point Imaginary line Real line Imaginary curve Category Projective geometry Elementary geometry stub ... more details
unreferenced date October 2011 File Transversal planes.JPG Plane t is a transversal plane because it intersects parallel planes p and q . thumb In geometry , a transversal plane is a plane that intersects not contains two or more Line geometry lines or Plane geometry planes . A transversal plane may also form dihedral angle s. Theorems Transversal plane theorem for lines Lines that intersect a transversal plane are parallel if and only if their alternate interior angles formed by the points of intersection are congruent. Transversal plane theorem for planes Planes intersected by a transversal plane are parallel if and only if their alternate interior dihedral angles are congruent. Transversal line containment theorem If a transversal line is contained in any plane other than the plane containing all the lines, then the plane is a transversal plane. Category Multi dimensional geometry Geometry stub ... more details
Lorentz curve may refer to the Cauchy Lorentz distribution , a probability distribution the Lorenz curve , a graphical representation of a different probability distribution disambig ... more details
Encyclopedia
top
