Methods for solving cubic equations appear in The Nine Chapters on the Mathematical Arta Chinese mathematical text compiled around the 2nd century BC and commented on by Liu Hui in the 3rd century.
Introduction In OpenGL, we can display a texture to small or large area. Displaying a texture in the same size requires copying all texels texture units to output pixels. If we want to create a large image from a small texture, OpenGL has to create many intermediate pixels.
This process of creating intermediate pixels is called as Interpolation.
OpenGL provides two interpolation methods to create a large texture image. In addition to OpenGL interpolation types, here different versions of Bi-Cubic interpolation are implemented in pixel shaders.
Bi-Cubic interpolation can provide good quality image on creating large image from small textures. Screenshot of ZoomInterpolation Application. Small area of flower is zoomed with different interpolations.
Different versions of interpolation have quality difference. Details of different interpolations are explained in the below sections. Background Interpolation is a "Process used to estimate an unknown value between two known values by utilizing a common mathematical relation".
ZoomInterpolation creates zoomed image with different interpolation types. OpenGL provides a Bi-Linear interpolation at its maximum. When zooming a very small texture area to large area, Bi-Linear provides a less quality.
Bi-Linear interpolation creates an intermediate pixel with the help of nearest 4 pixels. Bi-Cubic interpolation, a high quality version, creates an intermediate pixel with the help of nearest 16 pixels. In the nearest interpolation, intermediate pixels are created with the nearest valid pixel.
This kind of interpolated images are blocky since same data is copied to intermediate pixels. Edge of the flower seems to be a "stair case". We can remove this stair case through better interpolation methods.
We can use following code to achieve nearest interpolation in OpenGL fixed function pipeline. Therefore it is called Bi-Linear interpolation. Here "stair case" effect is not visible.
But we can see a shade of yellow and green through the edge of flower, and a small "stair case" is visible at some edges of the flower. When zooming a very small region of a texture, we can see some blocky edges in the output image. But we prepare a pixel shader with Bi-Linear interpolation.
We can improve the quality of Bi-Linear and get a better quality image. A pixel shader is a program which will execute for each pixels of a rendering.
More details of Pixel Shaders are available here or here. Bi-Linear Shader Implementation In Bi-Linear interpolation, the nearest four pixels are considered to create an intermediate pixel.
Texel is the term used to indicate an element in a texture similar to a pixel in screen [A texel, or texture element also texture pixel is the fundamental unit of texture space].
Initially two texels in top row and bottom row are interpolated with linear interpolation[horizontal]. Then interpolated pixels of top row and bottom row are also linearly interpolated[vertical].The middle one writes to the pipeline.
Write-Host and Out-Host writes to the console. 'echo' is an alias for Write-Output which writes to the pipeline as well. The best way to write to the console would be using the Write-Host cmdlet.. When an object is written to the pipeline it .
Edit Article How to Find Cubic Feet. In this Article: Article Summary Square or Rectangular Shape Cylindrical Shape Three-Sided Pyramid Four-Sided Pyramid Community Q&A Have you ever needed to fill a sandbox, a post hole, or any other three-dimensional space? Doing this requires “cubic measurement,” which is just another way of measuring volume. *The Greatest Integer Function, sometimes called the Step Function, returns the greatest integer less than or equal to a number (think of rounding down to an integer). In the mathematical subfield of numerical analysis, a B-spline, or basis spline, is a spline function that has minimal support with respect to a given degree, smoothness, and domain partition. Any spline function of given degree can be expressed as a linear combination of B-splines of that degree. Cardinal B-splines have knots that are equidistant from each other.
Hello Santhosh G_! Thank you for this great tutorial. I tried your bi-cubic interpolation method on different hardware and OpenGL versions.
On the NVIDIA Geforce GT with OpenGL and on the GTX with OpenGL Example 1: f is a cubic function given by f (x) = x 3.
Find the x and y intercepts of the graph of f. Find the domain and range of f. Sketch the graph of f. Solution to Example 1.
a - The y intercept is given by (0, f(0)) = (0, 0) The x coordinates of the x intercepts are the solutions to x . The quadratic function x^2 - 5x + 6 =0 can be factored into (x -3) * (x-2) And it would have roots of 3 and 2 So, if a cubic functions has roots of 2, 3 and 5 then its factors are/5(12).
Graphing Cubic Functions A step by step tutorial on how to determine the properties of the graph of cubic functions and graph them. Properties, of these functions, such as domain, range, x and y intercepts, zeros and factorization are used to graph this type of functions.
The "basic" cubic function, f (x) = x 3, is graphed below. The function of the coefficient a in the general equation is to make the graph "wider" or "skinnier", or to reflect it (if negative): The constant d in the equation is the y -intercept of the graph.