An Efficient Algorithm for Generating Slanted Ellipse Using Simultaneous Recurrences

Munetoshi NUMADA; Hiroyasu KOSHIMIZU; Yasuyo HATANO; Takayuki FUJIWARA; Takuma FUNAHASHI

doi:10.1587/transfun.E94.A.1458

An Efficient Algorithm for Generating Slanted Ellipse Using Simultaneous Recurrences

Munetoshi NUMADA, Hiroyasu KOSHIMIZU, Yasuyo HATANO, Takayuki FUJIWARA, Takuma FUNAHASHI

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Thus far, there have been many reports and publications on the algorithm for the efficient generation of a circle or an ellipse by the parametric method. In this parametric method, we compute a trigonometric function only at the time of setting the initial condition for generating graphics incrementally using the recurrence formula consisting of the arithmetical operations of addition, subtraction, and multiplication in the main loop. This means that the key to the faster generation of a circle or an ellipse is to reduce the number of multiplication operations. In the conventional methods, the numbers of multiplication operations required to generate a single point each for a circle and an ellipse are three and four, respectively. However, in this paper, we propose a method that makes it possible to generate a slanted ellipse by performing only two multiplication operations per point. The key to this is to use simultaneous recurrences. The proposed method allows a simpler initial setup than any of the conventional methods, thus performing the computation more efficiently. In addition, the new method proposed here causes no theoretical errors, with the rounding error being similar to or less than that of any conventional method.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E94-A No.6 pp.1458-1463

Publication Date: 2011/06/01

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Online ISSN: 1745-1337

DOI: 10.1587/transfun.E94.A.1458

Type of Manuscript: PAPER

Category: Computer Graphics

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Munetoshi NUMADA, Hiroyasu KOSHIMIZU, Yasuyo HATANO, Takayuki FUJIWARA, Takuma FUNAHASHI, "An Efficient Algorithm for Generating Slanted Ellipse Using Simultaneous Recurrences" in IEICE TRANSACTIONS on Fundamentals, vol. E94-A, no. 6, pp. 1458-1463, June 2011, doi: 10.1587/transfun.E94.A.1458.
Abstract: Thus far, there have been many reports and publications on the algorithm for the efficient generation of a circle or an ellipse by the parametric method. In this parametric method, we compute a trigonometric function only at the time of setting the initial condition for generating graphics incrementally using the recurrence formula consisting of the arithmetical operations of addition, subtraction, and multiplication in the main loop. This means that the key to the faster generation of a circle or an ellipse is to reduce the number of multiplication operations. In the conventional methods, the numbers of multiplication operations required to generate a single point each for a circle and an ellipse are three and four, respectively. However, in this paper, we propose a method that makes it possible to generate a slanted ellipse by performing only two multiplication operations per point. The key to this is to use simultaneous recurrences. The proposed method allows a simpler initial setup than any of the conventional methods, thus performing the computation more efficiently. In addition, the new method proposed here causes no theoretical errors, with the rounding error being similar to or less than that of any conventional method.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.E94.A.1458/_p

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@ARTICLE{e94-a_6_1458,
author={Munetoshi NUMADA, Hiroyasu KOSHIMIZU, Yasuyo HATANO, Takayuki FUJIWARA, Takuma FUNAHASHI, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={An Efficient Algorithm for Generating Slanted Ellipse Using Simultaneous Recurrences},
year={2011},
volume={E94-A},
number={6},
pages={1458-1463},
abstract={Thus far, there have been many reports and publications on the algorithm for the efficient generation of a circle or an ellipse by the parametric method. In this parametric method, we compute a trigonometric function only at the time of setting the initial condition for generating graphics incrementally using the recurrence formula consisting of the arithmetical operations of addition, subtraction, and multiplication in the main loop. This means that the key to the faster generation of a circle or an ellipse is to reduce the number of multiplication operations. In the conventional methods, the numbers of multiplication operations required to generate a single point each for a circle and an ellipse are three and four, respectively. However, in this paper, we propose a method that makes it possible to generate a slanted ellipse by performing only two multiplication operations per point. The key to this is to use simultaneous recurrences. The proposed method allows a simpler initial setup than any of the conventional methods, thus performing the computation more efficiently. In addition, the new method proposed here causes no theoretical errors, with the rounding error being similar to or less than that of any conventional method.},
keywords={},
doi={10.1587/transfun.E94.A.1458},
ISSN={1745-1337},
month={June},}

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TY - JOUR
TI - An Efficient Algorithm for Generating Slanted Ellipse Using Simultaneous Recurrences
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 1458
EP - 1463
AU - Munetoshi NUMADA
AU - Hiroyasu KOSHIMIZU
AU - Yasuyo HATANO
AU - Takayuki FUJIWARA
AU - Takuma FUNAHASHI
PY - 2011
DO - 10.1587/transfun.E94.A.1458
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E94-A
IS - 6
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - June 2011
AB - Thus far, there have been many reports and publications on the algorithm for the efficient generation of a circle or an ellipse by the parametric method. In this parametric method, we compute a trigonometric function only at the time of setting the initial condition for generating graphics incrementally using the recurrence formula consisting of the arithmetical operations of addition, subtraction, and multiplication in the main loop. This means that the key to the faster generation of a circle or an ellipse is to reduce the number of multiplication operations. In the conventional methods, the numbers of multiplication operations required to generate a single point each for a circle and an ellipse are three and four, respectively. However, in this paper, we propose a method that makes it possible to generate a slanted ellipse by performing only two multiplication operations per point. The key to this is to use simultaneous recurrences. The proposed method allows a simpler initial setup than any of the conventional methods, thus performing the computation more efficiently. In addition, the new method proposed here causes no theoretical errors, with the rounding error being similar to or less than that of any conventional method.
ER -