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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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Positive curves in polarized manifolds
Manuscripta Mathematica, 1997Curves \(Y\) on a smooth complex polarized variety \((X,A)\) of dimension \(k \geq 3\) are studied in order to generalize a paper by \textit{R. Paoletti} [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 6, No. 4, 259-274 (1995; Zbl 0874.14018)] in case \(k=3\) about the Seshadri constant \(\varepsilon (Y,A)\) and ...
Bădescu, Lucian +2 more
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Axial apodization using polar curves
SPIE Proceedings, 1991Two methods are described for shaping the irradiance distribution along the optical axis by the use of apodized polar curves as spatial filters.
Jorge Ojeda-Castaneda +1 more
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Curve Fitting in Polar Coordinates
The Journal of Experimental Education, 1951(1951). Curve Fitting in Polar Coordinates. The Journal of Experimental Education: Vol. 19, No. 3, pp. 267-269.
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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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STEADY STATE POLAR SENSITIVITY CURVES
GEOPHYSICS, 1939The resultant amplitude, “A,” of the combined outputs of a group of geophones is derived as a function of the difference in time of arrival, ΔT, at extreme geophones in the group—the number of geophones and the period of the waves being parameters.
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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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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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Curve sketching in polar coordinates
1983Open image in new window Fig. 3.1 In Fig. 3.1(a), the position of the point P is given by the position vector \( \overrightarrow {OP} \) or by the cartesian coordinates (x, y), where ON = x, NP = y, with the normal sign convention applying when P takes positions in the other quadrants defined by the axes Ox and Oy. The point P may also be given
H. M. Kenwood, C. Plumpton
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