Cylinder Reflections The Mathematics Behind the Images Cylinder Reflections The Mathematics Behind the Images Dr. Don Spickler Dr. Jennifer Bergner Salisbury University [email protected][email protected]Drs. Spickler & Bergner (Salisbury Univ.) Cylinder Reflections November 12, 2011 1 / 36
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Cylinder Re ections - Salisbury University · Cylinder Re ections The Mathematics Behind the Images Cylinder Re ections The Mathematics Behind the Images Dr. Don Spickler Dr. Jennifer
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Cylinder Reflections The Mathematics Behind the Images
Cylinder ReflectionsThe Mathematics Behind the Images
Anamorphic art is created by distorting animage so that is only revealed from a singlevantage point or from its reflection on amirrored surface. This artistic process wasfirst attempted during the Renaissance andbecame exceedingly popular during theVictorian Era.
The earliest known examples come from thenotebooks of Leonardo da Vinci. Hesuccessfully sketched an eyeball in 1485 thatcould only be discerned when looking at thedrawing from a certain angle.
As you can see, Julian Beever likes toincorporate human subjects in all of hissidewalk art. This shows that not only canhe create the perspective shift but he can doit to scale.
Hans Hamngren and Istvan Orosz usethe mirrored cylinder technique. Theyachieve this illusion by either drawingthe image on a distorted grid, similarto the way M. C. Escher createdmany of his illusions, or looking at themirrored image while drawing on aflat surface.
The Mathematics Step 4: Find the normal vector from intersection.
The Mathematics
Step 4: Find the normal vector fromintersection.
The normal vector is perpendicular to thesurface and is used in the calculation of thereflection line. For a cylinder, the normalvector will be parallel to the xy -plane andpass through the points 〈Ix , Iy , Iz〉 and〈0, 0, Iz〉. So our normal vector is thedifference between these two points,
The Mathematics Step 5: Find the reflection vector from intersection.
The Mathematics
Step 5: Find the reflection vector fromintersection.
This is probably the most involvedcalculation in the process. From the diagramon the right notice that the reflection vectorr = v + 2a where v = V − P is the vectorfrom the pixel to the viewer.
The Mathematics Step 6: Find the intersection of reflection line and paper.
The Mathematics
Step 6: Find the intersection ofreflection line and paper.
The paper is on the xy -plane so every threedimensional point on the paper has a zcoordinate of 0. We can use this fact to findhow far we must move down the reflectionvector until we hit the paper, this is thevalue of t in the reflection line formula,
Plot this point on the paper in the samecolor as the original pixel color and move onto the next pixel. When all of the points areplotted you have your transformed image.