Pegasus: A Drawing System for Rapid Geometric Design
Takeo Igarashi, 2Sachiko Kawachiya, 1Hidehiko Tanaka
1
Dept. of Info. Engneering, 2Dept. of Info. Science,Univ. of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan+81-3-3812-2111 ext 7413
{takeo, tanaka}@mtl.t.u-tokyo.ac.jp,
sachiko@is.s.u-tokyo.ac.jp

1

Satoshi Matsuoka
Dept. of Math. and Comp. Science,
Tokyo Inst, of Tech.
2-12-1 Ookayama, Meguro-ku.
Tokyo, Japan, +81-3-5734-3876
matsu@is.titech.ac.jp

ABSTRACT

Pegasus is a prototype drawing system for rapid geometric
design. Using Pegasus, a user can construct precise
geometric diagrams easily without struggling withcomplicated editing commands. Two novel interaction
techniques are introduced: interactive beautification andpredictive drawing. Interactive beautification receives theuser's free stroke input and beautifies it by considering
possible geometric constraints among segments bygenerating multiple candidates to prevent recognition
errors. Predictive drawing predicts the user's next drawing
operation based on the spatial relationship among existing
segments on the screen.

Figure 1: A diagram drawn on Pegasus: thisdiagram was drawn without any editing commands
such as rotation, duplication, or gridding.

Keywords

Geometric Design, Pen-based input, Drawing editor,
Predictive interface, Beautification.
INTRODUCTION

Construction of precise geometric diagrams is still a hard,time consuming task. Although commercial CAD and
Object-Oriented (OO) drawing systems provide various
editing operations, it is quite difficult to find the appropriate
combination of editing operations, especially for novice
users [2]. For example, the user must combine duplication,
flipping, and location adjustment operations to draw asymmetric diagram, and must use duplication and 90 degree
rotation to achieve perpendicularity.
To free the users of these complicated editing commands,
we propose two interaction techniques, interactive
beautification [3] and predictive drawing, and implement
them in a prototype system called Pegasus (PerceptuallyEnhanced Geometric Assistance Satisfies US!). The
combination of these two techniques enables even novice
users to draw complicated geometric diagrams (an example
is shown in Figure 1) rapidly without using any editing
commands.
INTERACTIVE BEAUTIFICATIONInteractive beautification is basically a free stroke
beautification with the flollowing added interactive

behavior. First, the user draws an approximate shape of his
desired segment with a free stroke using a pen or a mouse(Figure 2a). Then, the system infers geometric constraints
the input stroke should satisfy by checking the geometric
relationship among the input stroke and existing segments(Figure 2b). Finally, the system calculates the coordinatesof the beautified segment by solving the simultaneous
equations of inferred constraints, and displays the result tothe user (Figure 2c).
Technical problem is that, free stroke input is ambiguous innature and the beautified segment can frequently fail tocorrectly capture the user's intention. To solve the problem,
interactive beautification infers all possible candidates andallows the user to select one (Figure 2d). The selection isachieved by tapping on the intended candidate segment.
con n ectio na)b ) righ t an glec)d)align m en t

F ree stroke
input

In ferred
constraints

B eau tified
segm en t

M ultiple
candidates

Figure 2: Interactive beautification
PREDICTIVE DRAWINGGeometric illustrations often contain numerous identical
local configurations. Duplicate command is used inexisting drawing editors to generate identicalconfigurations. By contrast, by using interactive

beautification, duplication is implicitly achieved by drawing
a similar sketch.
Predictive drawing mechanism further assists theconstruction of identical configurations actively. If the userdraws an object whose shape is identical to some existing
segment, the system automatically predicts that the user
may draw similar segments in the vicinity of the newly
drawn segment. The predicted segments are displayed on
the screen, and the user can select one by tapping on it if it
happens to be the intended segment.
To be specific, the current predictive drawing mechanism
works as follows (Figure 3a):1.

When a new segment (trigger segment) is added to thescene as a result of interactive beautification, thesystem searches for the existing segments (referencesegments) whose length and angle are identical to thetrigger segment.

2.

The system records the spatial relationships among thereference segments and the segments (contextsegments) directly connected to the referencesegments.

3.

Finally, the system generates predicted segments in thevicinity of the trigger segment in such a way that therelation between each predicted segment and thetrigger segment is identical to that of the context
segment and the reference segment.

In addition, the system automatically supports theconstruction of symmetric diagrams and rotated diagrams
by adding flipped and rotated segments to the referencesegments (Figure 3b).
When the user selects a predicted candidate by tapping, thesystem immediately generates the next predictions using theselected candidate as a trigger segment. As a result, he canconstruct a diagram by successive tapping as long as
predicted segments contain the user's intended one. If theuser does not find any segments that are useful among thepredictions segments, he can seamlessly proceed to theinteractive beautification process by drawing the next freestroke.
mirrorrotation

Existing segments

New segment
and predictiona) Basic Prediction

Existing segments

New segment
and predictionb) Advanced Prediction

Trigger segment

Context segment

Reference segment

Predicted segment

Figure 3: Predictive drawing

RELATED WORK

Existing sketch beautification systems (Apple Newton,SmartSketch, etc.) do contain some features of interactive
beautification, but they support only primitive geometricconstraints such as connection, while Pegasus supports
more complex, global constraints such as symmetry andcongruence. Previous beautification systems [8][5] arebasically batch-based, which can cause too many errors
because of ambiguity in the user's input. Interactive
beautification avoids this problem by interactively
requesting the user's confirmation for each segment.
Automatic inference of geometric constraints can be seen insome research systems [7][4], but they mainly focus on thespecification of persistent spatial alignment among movable
objects, while we focus on static geometric relationships
among line segments in a drawing (parallelism, symmetry,etc.). Prediction mechanisms have been explored in several
research efforts [1][6], but their predictions are mainly
based on the regularity found in the operation sequence(repeated operation, etc.). Our prediction mechanism isdifferent in that we make use of regularities found in static
spatial configurations in a given drawing.
STATUS

The currently implemented constraints are: connection
between segments; horizontal or vertical alignment ofvertices; congruence, parallelism, and perpendicularity
among segments; horizontal and vertical symmetry; andequality of intervals between segments. Curves and arcs are
not supported yet but are being planned.
Pegasus is successfully working on small pen computers
and Xerox LiveBoards. User experiments showed that theusers can draw more rapidly and precisely with Pegasus
than with an existing CAD system and an OO drawing
editor.
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