The photo above is one in a series of artist books I've created. Rockbound Book: Canyon is made of slices of chert, laser-cut mat board pages, and bound with dyed viscose fabric. The first photo shows it in front of its hinged wooden box, 6" x 7" x 4". The second photo shows it in its box so you can see the rock slice that is its front cover. (Note the wooden box was routed by a different process.)
I wanted the pages to create a volumetric shape that morphs from the irregularly shaped front cover to an irregularly shaped back cover. I got interested in this category of small sculpture when I began binding pages into rock covers. My first few trials were unsuccessful because my page materials were too flimsy for the weight and rigidity of the rock covers. Then I hit upon using a laser cutter to cut mat board as my pages.
My next stumbling block was figuring how to achieve consistently successful interpolations from one shape to the other. While I used this technique for artist books, it's applicable to a wide variety of projects.
Step 1: Trace and Scan the Shapes
In my case, I wanted to create a volume made of separate, shaped pages between two slices of rock. Since the rock colors and shapes reminded me of the canyons of the American Southwest, I wanted the pages between the covers to slope down, like a canyon, so I created an intermediate middle shape. The interior pages morph into a central funneling channel like an arroyo, wash or slot canyon.
First I traced the front and back cover rock shapes using a black pen on white paper.
Then I scanned the tracings and converted the scans to pure black and white images using a threshold function in image editing software. I spent some time cleaning up the black-filled images, removing white speckles on the filled black shapes and black speckles from the white portions, so that if I were to use an auto-trace function in Inkscape I wouldn't get unnecessary holes, speckles or complexity.
Step 2: Trace the Back Cover Image
While you can use an auto-trace function (Inkscape: Path/Trace Bitmap/Color-Quantization=8), I have had best results using the Bezier curve tool to trace the shape myself. It results in a simpler and easier-to-manipulate path. I place the scanned image on a lower layer and use the Bezier tool to draw the path. I like to change the stroke paint color to red so it's easy to see against the black background image.
Step 3: Create the Shape to Interpolate To
This is the most important point. After trying a number of interpolations and getting bizarre and un-cuttable results, I realized that the most straightforward interpolation is if the beginning and ending shapes have the same number of points in similar relative positions. So instead of making a new tracing of the mid-point shape to interpolate to, I make an exact copy of my front cover path and edit into the shape I want to interpolate to.
I delete or hide the front cover scan layer.
I import and position the mid-point scan to most closely match the lower left point of the front cover path. I use the opacity control on the layer tool box to see that I've got the lower left corners best aligned. (Which corner you want to align will depend on the purpose of your interpolation. In this case, I knew I would bind the book along its left edge.)
Step 4: Edit the Copy of the Front Cover Path for the Mid-point and the Back Cover
Now I use the Edit Node tool to move the point or its Bezier handles so that each point in the new version of the path conforms to the edge of the mid-point shape. Then I do the same thing for the back cover shape. When I'm done, I hide or delete the layers with the scans on them.
I copy the mid-point shape so that I can use it twice - once with the front cover and once with the back cover.
I'm left with four aligned paths -- the front shape, two copies of the mid-point shape and the back cover shape.
Step 5: Interpolation
Now I can select the two shapes I want to interpolate between. I choose Extensions/Generate-from- Path/Interpolate... You will get a dialog box that allows you to choose how many intermediate steps you want and whether you want your end shapes repeated.
Ta-da! Beautifully topographic interpolated shapes that are eminently cuttable.
Step 6: Impose the File for Laser Cutting
The last step in my process is to separate out the shapes so they can be laser-cut. Inkscape makes this easy.
Once your interpolations are ungrouped (they appear as a group by default) and all shapes are selected, you can use the Object/Arrange menu for a dialog box that allows you to specify the number of rows and columns and the amount of space separating them.
Voila, an interpolation process that will give you usable results.