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- Barrera, Tony, et al.
(author)
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A chronological and mathematical overview of digital circle generation algorithms : Introducing efficient 4- and 8-connected circles
- 2016
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In: International Journal of Computer Mathematics. - : Informa UK Limited. - 0020-7160 .- 1029-0265. ; 93:8, s. 1241-1253
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Journal article (peer-reviewed)abstract
- Circles are one of the basic drawing primitives for computers and while the naive way of setting up an equation for drawing circles is simple, implementing it in an efficient way using integer arithmetic has resulted in quite a few different algorithms. We present a short chronological overview of the most important publications of such digital circle generation algorithms. Bresenham is often assumed to have invented the first all integer circle algorithm. However, there were other algorithms published before his first official publication, which did not use floating point operations. Furthermore, we present both a 4- and an 8-connected all integer algorithm. Both of them proceed without any multiplication, using just one addition per iteration to compute the decision variable, which makes them more efficient than previously published algorithms.
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- Barrera, Tony, et al.
(author)
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Connected Minimal Acceleration Trigonometric Curves
- 2005
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In: SIGRAD 2005 The Annual SIGRAD Conference Special Theme – Mobile Graphics November 23-24, 2005 Lund, Sweden.
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Conference paper (peer-reviewed)abstract
- We present a technique that can be used to obtain a series of connected minimal bending trigonometric splines that will intersect any number of predefined points in space. The minimal bending property is obtained by a least square minimization of the acceleration. Each curve segment between two consecutive points will be a trigonometric Hermite spline obtained from a Fourier series and its four first terms. The proposed method can be used for a number of points and predefined tangents. The tangent length will then be optimized to yield a minimal bending curve. We also show how both the tangent direction and length can be optimized to give as smooth curves as possible. It is also possible to obtain a closed loop of minimal bending curves. These types of curves can be useful tools for 3D modelling, etc.
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- Barrera, Tony, et al.
(author)
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Minimal Acceleration Hermite Curves
- 2005
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In: Game programming gems 5. - Hingham, Massachusetts : Charles River Media, inc.. - 1584503521 ; , s. 225-231
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Book chapter (pop. science, debate, etc.)
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| 8. |
- Barrera, Tony, et al.
(author)
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Minimal Acceleration Hermite Curves
- 2005
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In: Game Programming Gems 5. - : Charles River Media, Hingham, Massachusetts. - 1584503521 ; , s. 225-231
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Book chapter (peer-reviewed)abstract
- This gem shows how a curve with minimal acceleration can be obtained using Hermite splines [Hearn04]. Acceleration is higher in the bends and therefore this type of curve is a minimal bending curve. This type of curve can be useful for subdivision surfaces when it is required that the surface has this property, which assures that the surface is as smooth as possible. A similar approach for Bézier curves and subdivision can be found in [Overveld97]. It could also be very useful for camera movements [Vlachos01] since it allows that both the position and the direction of the camera can be set for the curve. Moreover, we show how several such curves can be connected in order to achieve continuity between the curve segments.
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- Barrera, Tony, et al.
(author)
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Trigonometric splines
- 2008. - 1
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In: Game programming Gems 7. - Boston : Charles River Media. - 9781584505273 - 1584505273 ; , s. 191-198
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Book chapter (pop. science, debate, etc.)
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| 10. |
- Barrera, Tony, et al.
(author)
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Vectorized table driven algorithms for double precision elementary functions using Taylor expansions
- 2009
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In: APLIMAT 8th international conference. ; 2, s. 171-178
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Conference paper (peer-reviewed)abstract
- This paper presents fast implementations of the inverse square root and arcsine, both in double precision. In single precision it is often possible to use a small table and one ordinary Newton-Raphson iteration to compute elementary functions such as the square root. In double precision a substantially larger table is necessary to obtain the desired precision, or, if a smaller table is used, the additional Newton-Raphson iterations required to obtain the precision often requires the evaluation of other expensive elementary functions. Furthermore, large tables use a lot of the cash memory that should have been used for the application code.Obtaining the desired precision using a small table can instead be realised by using a higher order method than the second order Newton-Raphson method. A generalization of Newton's method to higher order is Householder's method, which unfortunately often results in very complicated expressions requiring many multiplications, additions, and even divisions.We show how a high-order method can be used, which only requires a few extra additions and multiplications for each degree of higher order. The method starts from the Taylor expansion of the difference of the value of the elementary function and a starting guess value for each iteration. If the Taylor series is truncated after the second term, ordinary Newton iterations are obtained. In several cases it is possible to algebraically simplify the difference between the true value and the starting guess value. In those cases we show that it is advantageous to use the Taylor series to higher order to obtain the fast convergent method. Moreover, we will show how the coefficients of a Chebyshev polynomial can be fitted to give as little error as possible for the functions close to zero and in the same time reduce the terms in the Taylor expansion.In the paper we benchmark two example implementations of the method on the x86_64 architecture. The first is the inverse square root, where the actual table (to 12 bit precision) is provided by the processor hardware. The inverse square root is important in many application programs, including computer graphics, and explicit particle simulation codes, for instance the Monte Carlo and Molecular Dynamics methods of statistical mechanics. The other example is the arcsine function, which has a slow converging Taylor expansion and where no tables are provided by the hardware. The vectorized versions of the implementations of the inverse square root are 3.5 times faster than compiled code on the Athlon64 and about 5 times faster on the Core 2. The scalar version of the arcsine function is, depending on order and table size, between 2 and 3 times faster than the compiled code, and the vectorized version is between 3 and 4 times faster on the Athlon64, while it is between 4 and 5 times faster than the compiled version on the Core 2.
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