system with children in an educational TV program in Japan and found that even elementary-school students could quickly generate reasonably interesting animations.

(a) Rest shape
(b) setting handles
(c) Rotation with two handles
(d) stretching with two handles
(e) Deformation with three handles
figure 7. as-rigid-as-possible shape manipulation. users place handles on the drawing, then
manipulate it by moving the handles.

 

shape deforms naturally in response to user input; for users it feels like they’re manipulating a physical object.

Traditional computer-based methods for shape manipulation are roughly divided into three categories:

Skeleton. 13 The user embeds a skel-etal structure inside the shape and controls it to deform the shape. However, embedding a skeleton in each shape is tedious, and the approach does not work for stretching and squashing;

Spatial deformation. 14 The user de-
fines a spatial mapping using several
control points, then deforms the shape
according to the mapping function.
However, mapping functions do not
consider the rigidity of the shape and
result in unnatural deformation; and
Our method takes a completely geo-
metric approach, defining an energy
function that measures the amount
of geometric deformation, then mini-
mizes it using an optimization method.
We designed the energy to give a closed-
form solution to the problem. In it, the
system obtains the deformation by
solving two large sparse linear-matrix
equations in sequence, a very fast and
perfectly stable approach.

It is also particularly useful for creating 2D animations. Traditional animation artists assemble many slightly different drawings to create an animation. In our shape-manipulation system MovingSketch ( http://www-ui.is.s.u-tokyo.ac.jp/~takeo/research/rigid/mov-ingsketch/ index.html), users create an interesting animation by drawing a character and recording the manipulation process. Using a multi-touch input device, 16 they grab a character with both hands and manipulate it to create an animation (see Figure 8). We tested the

Lessons Learned

Each of these projects addresses a specific problem, with technical contributions being rather independent. However, emerging from them are common guidelines for designing a compelling user experience:

Natural to humans. First, start with what is natural to a human rather than with what is natural to a computer. The computer represents a 3D model with a collection of 3D points and their connections; traditional computer-aided-design systems ask users to provide this information directly. Advanced systems represent a model with a sequence of editing operations, but most of them still require that users be aware of points and faces. Similarly, a computer represents a 2D drawing with its position and orientation. Traditional drawing systems ask users to directly control these parameters; that is, traditional systems expose the underlying representation to the user directly. Though it is the most straightforward way to implement a system, the result means difficulty for novice users. That’s why we start by identifying the most natural operations for a human referring to real-world examples, then developing an algorithm that maps