Advanced computer graphics techniques such as ray tracing are being used to visualize complicated three-dimensional surfaces. However, any single image of such a surface can only provide a partial representation of it, since part of the surface is always blocked from view. Multiple images from several different viewpoints can be used to cover the entire surface, however mentally integrating the images to understand of the total surface can be difficult when viewpoints are not correlated in a natural way.

An interesting alternative to using multiple images for visualization is to place synthetic mirrors in the scene with the surface to be visualized, positioned so as to reflect images of the non-visible portions of the surface back to the viewpoint. We get the advantages of multiple viewpoints while maintaining a conception of where these viewpoints come from.

To best realize these advantages, we must carefully position the mirrors in the scene so as to "optimally" view the surface. In this paper, we provide algorithms and geometries providing optimal mirror placements for two natural classes of problems; object-independent and object-dependent mirror placement.

In the object-independent formulation, we seek to design a "camera" with a fixed geometry of mirrors and viewpoint which satisfies certain criteria for all objects in a specific class. The object-independent formulation is well-suited to visualizing scientific data sets, since extensive analysis prior to an initial rendering may be expensive or (as in the case of volumetric data without an explicit geometric model) impossible. we present the design of a camera which guarantees complete visibility for any orientation of any convex object in both 2 and 3 dimensions using the minimum number of mirrors, and which also minimizes the maximum distance between a point in the object and its (virtual) viewpoint. This distance criteria is important because it ensures that the reflected images are as high-resolution as possible.

Here we present ray-traced images showing simulated views from our optimal mirrors configuration. We literally provide a new way to see the world. An full version of our paper is available in postscript, as well as a journal submission in progress.

Here and above, we visualize the Earth using mirrors. Note the difference between the viewpoints of the two impages. All shadows and light reflections have been eliminated, resulting in a "flat" image, like a map. The central disk shows the part of the earth that can be directly seen from the viewpoint, while the three elliptic disks are the mirrors that show the inverted images of three caps. Notice that the Earth is completely seen.

An image which shows three different views through the three mirrors. Even if the object to be seen is not convex, we think that this image shows the potentiality of our "camera": it allows to clearly see and understand the scene in a very complete way. (The cover is not completely seen because it is placed outside the "camera".)

Visualizing a convex body using mirrors. Our "camera" allows to completely visualize any convex object positioned in the interior of the sphere by using the three mirrors. Note the hidden surfaces revealed by the mirror.

These images were generated using rayshade. Thanks to Belén Palop and Martí Sanchez, of the Dep. Matemàtica Aplicada II of the Universitat Politècnica de Catalunya.