tag:blogger.com,1999:blog-1177691919768241768.post1952610381083227409..comments2017-04-16T23:00:52.484-07:00Comments on Ben's blog: Affine transformations and their inverseBenhttp://www.blogger.com/profile/16266302090312287528noreply@blogger.comBlogger5125tag:blogger.com,1999:blog-1177691919768241768.post-14252798144241961942016-01-31T00:31:10.426-08:002016-01-31T00:31:10.426-08:00This comment has been removed by the author.Bleak Disclosurehttp://www.blogger.com/profile/03420066873113447711noreply@blogger.comtag:blogger.com,1999:blog-1177691919768241768.post-67696751678186291212016-01-31T00:30:40.445-08:002016-01-31T00:30:40.445-08:00In the more general case of an arbitrary 2D or 3D ...In the more general case of an arbitrary 2D or 3D affine transform, I would just use formulas 3 and 6 from <a href="http://mathworld.wolfram.com/MatrixInverse.html" rel="nofollow">MathWorld : Matrix Inverse</a> for the "R" matrix.<br /><br />If the determinant is zero, you have no inverse. Be aware of stability concerns and finite precision.Bleak Disclosurehttp://www.blogger.com/profile/03420066873113447711noreply@blogger.comtag:blogger.com,1999:blog-1177691919768241768.post-26910251033530123492016-01-31T00:04:25.404-08:002016-01-31T00:04:25.404-08:00Thanks Ben, this would have saved me a lot of time...Thanks Ben, this would have saved me a lot of time back in the day. I hate to be this guy, but...<br /><br />Although @dgk is right as far as it goes, the last missing component is a shear transformation (Assuming we're discussing 2D, I am not sure about higher dimensions). This gives us the sixth degree of freedom, assuming the scaling mentioned earlier is non-uniform.<br /><br />With scaling, it's possible for the transform to be non-invertible, giving infinite solutions. With non-uniform scaling or shearing, the quality of an invertible transformation can become unstable (<a href="https://en.wikipedia.org/wiki/Condition_number" rel="nofollow">Condition Number</a>). This makes the matrix inversion a bit more complicated.<br /><br />Thanks Manoj for your insightful <a href="http://mathworld.wolfram.com/AffineTransformation.html" rel="nofollow">MathWorld</a> quote.<br />Bleak Disclosurehttp://www.blogger.com/profile/03420066873113447711noreply@blogger.comtag:blogger.com,1999:blog-1177691919768241768.post-6898642055481108792014-04-10T05:11:07.935-07:002014-04-10T05:11:07.935-07:00Although this is right as far as it goes, the inve...Although this is right as far as it goes, the inversion only covers a rotation+translation transform, not a (rotation+scale)+translation transform, where the R-1 needs to be scaled by it's determinant, which is not unity for a scale-change. This makes the inverse matrix a bit more complicated :-)dgkhttp://www.blogger.com/profile/02742769742717528300noreply@blogger.comtag:blogger.com,1999:blog-1177691919768241768.post-3052933174589496442012-09-03T02:25:21.860-07:002012-09-03T02:25:21.860-07:00You have very intellectual to express anything as ...You have very intellectual to express anything as you discussed affine transformation, an affine transformation is any transformation that preserves col-linearity and ratios of distances. In this sense, affine indicates a special class of projective transformations that do not move any objects from the affine space to the plane at infinity or conversely. An affine transformation is also called an affinity.<br /><a href="http://www.tutorcircle.com/how-to-construct-a-trapezium-flr1q.html" rel="nofollow">How to Construct a Trapezium</a><br />manoj singhhttp://www.blogger.com/profile/16276013343106341394noreply@blogger.com