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2000
Polar curves with respect to proper conics and polar surfaces with respect to proper quadrics are investigated. Polar curves (surfaces) are defined as the envelope of the polar lines (planes) of the points on a given curve (surface) with respect to a quadratic curve (surface).
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Polar curves with respect to proper conics and polar surfaces with respect to proper quadrics are investigated. Polar curves (surfaces) are defined as the envelope of the polar lines (planes) of the points on a given curve (surface) with respect to a quadratic curve (surface).
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1984
As everybody knows, if C is a plane curve, 0 is a point of C of multiplicity \(\mu\), and C' is a polar curve of C, then C' passes through 0 with multiplicity \(\mu\) '\(\geq \mu -1\). In the present paper it is pointed out that, whatever the characteristic of the base field, if C is reduced it is not true, in general, that a polar curve C' passes with
BECCARI G., MASSAZA, Carla
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As everybody knows, if C is a plane curve, 0 is a point of C of multiplicity \(\mu\), and C' is a polar curve of C, then C' passes through 0 with multiplicity \(\mu\) '\(\geq \mu -1\). In the present paper it is pointed out that, whatever the characteristic of the base field, if C is reduced it is not true, in general, that a polar curve C' passes with
BECCARI G., MASSAZA, Carla
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The Polar Equations of a Curve
The Mathematical Gazette, 1931It is well known that a curve has only one equation in Cartesian co-ordinates When referred to a given pair of lines as axes, but it does not appear to be so well known that a curve can have many equations in polar co-ordinates, all referred to the same pole and initial line.
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The Mathematics Teacher, 2013
Graphing the polar function on a rectangular plane simplifies graphing, increases student understanding, and reinforces connections.
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Graphing the polar function on a rectangular plane simplifies graphing, increases student understanding, and reinforces connections.
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1982
As we mentioned in the Introduction to Chapter 9, the calculation of lengths of curved lines was one of the principal problems that led to the creation of the calculus. It was an old and intractable problem. Archimedes had used polygons inscribed in a circle to calculate π, but nothing further was discovered about curve lengths until the seventeenth ...
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As we mentioned in the Introduction to Chapter 9, the calculation of lengths of curved lines was one of the principal problems that led to the creation of the calculus. It was an old and intractable problem. Archimedes had used polygons inscribed in a circle to calculate π, but nothing further was discovered about curve lengths until the seventeenth ...
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Single-valued curves in polar coordinates
Computer-Aided Design, 1990Single-valued Bézier curves are of the form \(y=f(x)\) and can only model a very limited variety of shapes. This paper treats such functions in polar coordinates, in the form \(r=r(0)\). The author derives recursive algorithms and shows that the corresponding basis functions are expressed in terms of sines and cosines.
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2000
Section 6.1 is a short introduction to using polar coordinates with MAPLE. In Sections 6.2–6.5 we plot the polar graphs of some remarkable curves (in particular, spirals, roses and crosses) and use inversion transformation.
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Section 6.1 is a short introduction to using polar coordinates with MAPLE. In Sections 6.2–6.5 we plot the polar graphs of some remarkable curves (in particular, spirals, roses and crosses) and use inversion transformation.
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Exploroing Polar Curves with GeoGebra
The Mathematics Teacher, 2012Most trigonometry textbooks teach the graphing of polar equations as a two-step process: (1) plot the points corresponding to values of θ such as π, π/2, π/3, π/4, π/6, and so on; and then (2) connect these points with a curve that follows the behavior of the trigonometric function in the Cartesian plane. Many students have difficulty using this method
Tuyetdong Phan-Yamada, Walter M. Yamada
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The polar confidence curve for a ratio
Econometric Reviews, 2019AbstractBased on Fieller’s method for the estimation of a confidence set for a ratio, I construct a polar plot of the test statistics for all angles associated with the ratio.
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A mechanical model for polar curves
European Journal of PhysicsAbstract In this work we discuss the feasibility of the well-known classical polar curves (limaçons, roses and logarithmic spirals), studied in calculus courses, to emerge naturally as trajectories of a mass-spring system, limited to moving along a passing axis that rotates in uniform circular motion with controlled angular velocity ...
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