How can you prevent floating point errors in computational geometry algorithms?

See, every algorithm you write needs to be generic first of all and should not be affected by the type of data we are traversing on. And if you are making changes based on a specific data type like set of decimals then we should be handing that case earlier

Understanding the performance constraints v/s precision requirements should be noted for such cases - this is the basis of determining which libraries to use. It's important to note that using exact arithmetic will affect the performance - making it slow, memory-intensive for large numbers.

When comparing two floating point datasets for common elements (e.g. intersecting point sets) I make the comparison based on rounded copies of each elements while being sure to keep the original unrounded data to preserve its floating point accuracy.

"Easy, just use infinite-precision arithmetic!" For awareness for non-mathematicians, this does not imply "big data". "Infinite-precision arithmetic" is just a code phrase for: keeping numbers whole, I.e. not chopping bits of numbers off and discarding them. Infinite-precision (i.e. exact arithmetic) uses original "intact" numbers without any rounding. If performance issues arise, "variable precision" and "arbitrary precision" methods are ways to perform arithmetic where the user determines / controls loss of precision. These methods can be automated to optimally balance complexity, with performance e.g. in environments with "surge velocity" of data. cheers

In computational geometry, precision is paramount. Floating point errors, a common challenge, can distort results, especially in intricate calculations. To counter this, employing exact arithmetic is crucial. Instead of floating points, exact numbers like integers or rational numbers (fractions) ensure accuracy. For example, using coordinates as fractions (like 1/3 instead of 0.3333...) eliminates rounding errors. This approach is essential in algorithms where precision dictates reliability, such as in computational topology, GIS applications, and computer-aided design. By leveraging exact arithmetic, we ensure robust, error-free computational geometry solutions.

Robust predicates are crucial in computational geometry, ensuring accurate spatial relationships between geometric entities. These predicates, designed to handle nearly degenerate scenarios, minimize errors in tasks like point-in-polygon tests and convex hull computations. Addressing limitations of finite precision arithmetic, robust predicates employ advanced mathematical techniques, such as Shewchuk's work, providing consistent and reliable results. Integrating them into algorithms enhances overall reliability, mitigating numerical instability and ensuring accurate outcomes across diverse geometric scenarios. Their computational overhead is generally justified by the precision and correctness they bring to geometric computations.

Precision and performance is again at odds. A simple trick to ensure precision is as said in the definition, use symboling arithmetic, Well i have an easy but not so sustainable solution, string comparison . Instead of comparing floats if the precision is with in reasonable number of digits it is an option we convert the numbers to string and a precise matching follows, however this is not sustainable even with hardware accelerated string comparison methods for heavy computations.

Robust predicates, essentially well-defined mathematical conditions, are designed to counter floating point errors. Unlike standard floating-point arithmetic, which can introduce rounding errors. They are particularly vital in algorithms for geometric constructions and intersections, where precision is non-negotiable. By incorporating techniques like adaptive precision arithmetic or exact arithmetic computations, these predicates provide a safeguard against the inaccuracies that floating points can introduce. Implementing robust predicates is a best practice in computational geometry, ensuring reliable and precise algorithmic outcomes, crucial for applications ranging from computer graphics to spatial analysis.

I strongly suggest to either use strategy one (exact arithmetic) or robust predicates. A good example is to take a look at the LEDA platform for combinatorial and geometric computing. It implements both strategies (for example a rational number type).

Both exact arithmatic and robust predicates are promising strategies to counter floating-point errors in geometric computations. However, challenges, including heightened design complexity, amplify the burden on verification to ensure reliability across diverse geometric scenarios. Addressing these challenges is crucial to leveraging benefits without compromising performance since floating-point filters, aiding error detection, might trigger a shift to exact arithmetic due to detected inaccuracies, impacting computational efficiency. For precision-critical geometric operations, considering the CGAL library coupled with GNU MPFR for enhances numerical precision, form a robust basis for precise geometric computations.

The approach to use snap and round is similar to using fixed point arithmetic. Fixed point arithmetic is very predictable in terms of accuracies. Nowadays floating point is used often for problems where a "fractional representation" is required or people safe time to determine min/max numeric ranges ignoring the problems floating point introduces. Languages like C could have helped there by introducing fractional datatypes mapping on machine level only to integers, however they were never introduced and people get scared from using integer based fixed point math. Thir article should also mention fixed point math as a solution.

Snapping and rounding are vital precision management techniques in computational geometry. Snapping adjusts points to a grid, reducing small variations and enhancing numerical stability, useful in non-critical scenarios like visualization. Rounding involves limiting coordinate precision, mitigating cumulative errors during computations. Integrating these techniques improves algorithmic robustness, balancing efficiency with acceptable accuracy. In scenarios where exact precision isn't crucial, snapping and rounding prevent inaccuracies and contribute to reliable geometric results.

Rounding everything to an integer is the best way to eliminate floating-point errors.

Most problems map correctly regarding rounding and range when using double precision floating point in stead of 32 bit single precision floats.

Snapping refers to adjusting points to a predefined grid, which reduces errors caused by minute differences in calculations. Rounding, on the other hand, involves adjusting numbers to a fixed number of decimal places, ensuring consistency in arithmetic operations. These methods are particularly beneficial in tasks involving spatial data, like GIS mapping and 3D modeling. By implementing snapping and rounding, algorithms gain stability and robustness, crucial in applications.

Perturbation and simulation are useful when trying to validate a deterministic system, but not all floating point calculations are or need to be deterministic. Non-determinism in floating point algorithms is a key discussion point before creating the algorithm as that will influence where it fits into its ecosystem as well as how it's tested and validated

Perturbation involves slightly altering input data, which helps in avoiding degenerate cases often prone to calculation errors. This technique is crucial in algorithms dealing with sensitive geometric configurations, where even minor inaccuracies can lead to incorrect results. Simulation of simplicity, another key approach, treats floating point numbers as approximations of rational numbers, simulating exact arithmetic. This method allows for more accurate geometric computations, especially in complex algorithms. Both perturbation and simulation enhance the robustness of computational geometry algorithms, ensuring more reliable and error-resistant outcomes in fields like computer graphics and spatial data analysis.

Integrate perturbation techniques by slightly altering input data to observe algorithmic sensitivity and uncover potential instability. Simulate scenarios with varied inputs to assess algorithm robustness under diverse conditions, providing insights into floating-point error impact. This iterative process aids in refining algorithms, enhancing their resilience against numerical artifacts and improving overall computational geometry reliability.

Using error analysis is the best way to achieve this as it contributes to get the data on error cases like how much is the accuracy and how much percentage is without error. Considering the example of Wake word algorithms in which we can identify how much it has false positives.

Error analysis involves assessing and quantifying potential errors introduced by floating points. This preemptive approach allows developers to understand and mitigate inaccuracies in geometric computations. Validation, on the other hand, checks the correctness of outcomes, ensuring that the results align with geometric principles and expected accuracy. Together, these methods play a vital role in applications like 3D modeling and GIS, where precision is non-negotiable. Implementing error analysis and validation in computational geometry algorithms not only enhances accuracy but also ensures reliability and trustworthiness in outputs, a critical aspect in fields where decisions are based on computational results.

A strong and optimized validation process can help enhance the efficiency of the floating point error calculation by gradually narrowing down to the error source and evaluating it's magnitude. Machine learning techniques can be adapted for this method as well.

Using numerical analysis one can quantity error and variance. Simply randomizing the input seed to your RNG and logging important result or intermediate calculations can identify if your implementation is susceptible to accumulated roundoff error, and give an idea of how volatile a deviation there may be. Swapping out data types is a common fix, and can help diagnose how sensitive specific parts of the code are. Many powerful mitigations such as Kahan summation can be drop-in replacements in higher level languages.

One way to avoid accumulation of errors that is not mentioned here is to try and refactor the code in a way that avoids intermediate object which are not actually used for the resulting quantity you want to get at. That is to say, reflect on the code before you and consider whether there is a more direct way to get from in- to desired output.

To prevent floating-point errors in computational geometry algorithms, use robust geometric predicates and algorithms designed to minimize numerical instability. Additionally, consider employing exact arithmetic libraries or techniques like interval arithmetic to enhance precision and reduce the impact of rounding errors. Few of the techniques that can be worthy of it are: 1. Orientation Predicates: arbitrary precision arithmetic for computing determinants in orientation tests. 2. Robust Line Segment Intersection 3. Geometric Primitives with Exact Arithmetic 4. Snap Rounding: 5. Incremental Construction 6. Extended Precision 7. Adaptive Precision 8. Interval Arithmetic

We could use a precision level if we know error tolerance from the beginning. Thus, if the error tolerance should be till 3th digits we could use 10th or something to increase the chance of preventing errors.

If you can't use any external proven floating-point operation libraries, write your comparisons as mathematical epsilon neighborhoods and support these comparisons by writing unit tests according to your precision requirements, may be with a little higher precision than your requirements.

A key point to remember is the capabilities and limitations of the hardware your floating point algorithms run on. Ensuring you have proper floating point support on the hardware is key, because not all hardware has equivalent support. In the same way, not all compilers provide the same types of floating point support. As another example, the LLVM, GCC, and MSVC compilers offer equivalent floating point support, but it's up to the engineer to understand the difference between floating point layouts, IEEE support, and other various nuances to floating point arithmetic.