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Lecture No.11 2D Transformations I In the previous lectures so far we have discussed output primitive as well as filling primitives. With the help of them we can draw an attractive 2D drawing but that will be static whereas in most of the cases we require moving pictures for example games, animation, and different model; where we show certain objects moving or rotating or changing their size. Therefore, changes in orientation that is displacement, rotation or change in size is called geometric transformation. Here, we have certain basic transformations and some special transformation. We start with basic transformation.
Basic Transformations Translation Rotation Scaling Above are three basic transformations. Where translation is independent of others whereas rotation and scaling depends on translation in most of cases. We will see how in their respective sections but here we will start with translation. Translation A translation is displacement from original place. This displacement happens to be along a straight line; where two distances involves one is along x-axis that is tx and second is along y-axis that is ty.
The same is shown in the figure also we can express it with following equation as well as by matrix: x = x + tx, y = y + ty Here (tx, ty) is translation vector or shift vector. We can express above equations as a single matrix equation by using column vectors to represent coordinate positions and the translation vector: P = P + T Where P = P= T = Translation is a rigid-body transformation that moves objects without deformation. That is, every point on the object is translated by the same amount. A straight line can be translated by applying the above transformation equation to each of the line endpoints and redrawing the line between the new coordinates. Similarly a docsity.com. Polygon can be translated by applying the above transformation equation to each vertices of the polygon and redrawing the polygon with new coordinates.
Similarly curved objects can be translated. For example to translate circle or ellipse, we translate the center point and redraw the same using new center point. Rotation A two dimensional rotation is applied to an object by repositioning it along a circular path in the xy plane.
To rotate a point, its coordinates and rotation angle is required. Rotation is performed around a fixed point called pivot point. In start we will assume pivot point to be the origin or in other words we will find rotation equations for the rotation of object with respect to origin, however later we will see if we change our pivot point what should be done with the same equations.
Another thing is to be noted that for a positive angle the rotation will be anti-clockwise where for negative angle rotation will be clockwise. Now for the rotation around the origin as shown in the above figure we required original position/ coordinates which in our case is P(x,y) and rotation angle . When coordinate positions are represented as row vectors instead of column vectors, the matrix product in rotation equation is transposed so that the transformed row coordinate vector x,y is calculated as: P T = (R.
How to install ulaunchelf on ps2. RT Where PT and the other transpose matrix can be obtained by interchanging rows and columns. Also, for rotation matrix, the transpose is obtained by simply changing the sign of the sine terms.
Rotation about an Arbitrary Pivot Point: As we discussed above that pivot point may be any point as shown in the above figure, however for the sake of simplicity we assume above that pivot point is at origin. Anyhow, the situation can be dealt easily as we have equations of rotation with respect to origin. We can simply involve another transformation already read that is translation so simply translate pivot point to origin. By translation, now points will make angle with origin, therefore apply the same rotation equations and what next? Simply retranslate the pivot point to its original position that is if we subtract xr,yr now add them therefore we get following equations: x = xr + (x - xr) cos – (y - yr) sin y = yr + (x - xr) sin – (y - yr) cos As it is discussed in translation rotation is also rigid-body transformation that moves the object along a circular path.
Now if we want to rotate a point we already achieved it. But what if we want to move a line along its one end point very simple treat that end point as pivot point and perform rotation on the other end point as discussed above. Similarly we can rotate any polygon with taking some pivot point and recalculating vertices and then redrawing the polygon.
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Scaling A scaling transformation changes the size of an object. Scaling may be in any terms means either increasing the original size or decreasing the original size. An exemplary scaling is shown in the above figure where scaling factors used Sx=3 and Sy=2. So, what are these scaling factors and how they work very simple, simply we multiply each coordinate with its respective scaling factor. Therefore, scaling with respect to origin is achieved by multiplying x coordinate with factor Sx and y coordinate with Sx=3 Sy=2 docsity.com. Therefore, following equations can be expressed: x = x.Sx y = y.Sy In matrix form it can be expressed as: P = S.P Now we may have different values for scaling factor.
Therefore, as it is multiplying factor therefore, if we have scaling factor 1 then the object size will be increased than original size; whereas; in reverse case that is scaling factor.
Unsupported Browser We have detected that you are using Internet Explorer 6, a browser version that is not supported by this website. Internet Explorer 6 was released in August of 2001, and the latest version of IE6 was released in August of 2004. It is no longer supported by Microsoft. Continuing to run IE6 leaves you open to any and all security vulnerabilities discovered since that date. In March of 2011, Microsoft released version 9 of Internet Explorer that, in addition to providing greater security, is faster and more standards compliant than versions 6, 7, and 8 that came before it. Asus radeon r7 240 driver. We suggest installing the, or the latest version of these other popular browsers:,.
Unsupported Browser We have detected that you are using Internet Explorer 6, a browser version that is not supported by this website. Internet Explorer 6 was released in August of 2001, and the latest version of IE6 was released in August of 2004. It is no longer supported by Microsoft.
Continuing to run IE6 leaves you open to any and all security vulnerabilities discovered since that date. In March of 2011, Microsoft released version 9 of Internet Explorer that, in addition to providing greater security, is faster and more standards compliant than versions 6, 7, and 8 that came before it. We suggest installing the, or the latest version of these other popular browsers:,.
Unsupported Browser We have detected that you are using Internet Explorer 6, a browser version that is not supported by this website. Internet Explorer 6 was released in August of 2001, and the latest version of IE6 was released in August of 2004. It is no longer supported by Microsoft. Continuing to run IE6 leaves you open to any and all security vulnerabilities discovered since that date. In March of 2011, Microsoft released version 9 of Internet Explorer that, in addition to providing greater security, is faster and more standards compliant than versions 6, 7, and 8 that came before it.
We suggest installing the, or the latest version of these other popular browsers:,.
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