SAN FRANCISCO JULY 22-26
Volume 19, Number 3, 1985
I
Animating
Rotation
with
Quaternion
Curves
Ken Shoemaker
The Singer C o m p a n y
Link Flight Simulation Division
ABSTRA CT
Solid bodies roll and t u m b l e through space.
In
c o m p u t e r a n i m a t i o n , so do cameras. The r o t a t i o n s of
these objects are best described using a four coordinate
system, quaternions, as is shown in this paper. Of all
quaternions, those on the unit sphere are most suitable
for a n i m a t i o n , but the question of how to construct
curves on spheres has not been much explored. This
p a p e r gives one answer by presenting a new kind of
spline curve, c r e a t e d on a sphere, suitable for smoothly
in-hetweening (i.e. i n t e r p o l a t i n g ) sequences of a r b i t r a r y
rotations. Both t h e o r y a n d experiment show t h a t the
motion g e n e r a t e d is smooth and n a t u r a l , w i t h o u t quirks
found in earlier methods.
C.R.
Classification:
G . l . 1 [Numerical Analysis]
Interpolation--Spline
and
piecewise
polynomial
interpolation;
G.1.2
[Numerical
Analysis]
Approximation--Spline
and
piecewise
polynomial
a p p r o x i m a t i o n ; 1.2.9 [Artificial Intelligence] R o b o t i c s - M a n i p u l a t o r s ; 1.3.5 [Computer Graphics] C o m p u t a t i o n a l
G e o m e t r y and O b j e c t M o d e l l i n g - - C u r v e , surface, solid,
and object representation, - - G e o m e t r i c algorithms,
languages, and systems, - - H i e r a r c h y a n d geometric
transformations
General
Terms:
Algorithms, Theory
phrases:
quaternlon,
rotation,
spherical geometry, spline, B6zier curve, B-spline,
a n i m a t i o n , interpolation, a p p r o x i m a t i o n , in-betweening
ieywords
and
1. I n t r o d u c t i o n
C o m p u t e r a n i m a t i o n of three dimensional objects
i m i t a t e s the key .frame techniques of t r a d i t i o n a l
a n i m a t i o n , using key positions in space i n s t e a d of key
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©
1985 A C M 0-89791-166-0/85/007/0245
$00.75
drawings. Physics says t h a t the general position of a
rigid b o d y can be given b y combining a t r a n s l a t i o n with
a
rotation.
Computer
animators
key
such
t r a n s f o r m a t i o n s to control both s i m u l a t e d cameras a n d
objects to be rendered. In following such an a p p r o a c h ,
one is n a t u r a l l y led to ask: W h a t is the best
r e p r e s e n t a t i o n for general rotations, and how does one
in-between them? Surprisingly little has been published
on these topics, and the answers are not trivial.
This p a p e r suggests t h a t the common solution, using
three Euler's angles i n t e r p o l a t e d independently, is not
ideal. T h e more recent (1843) n o t a t i o n of quaternions
is proposed instead, along with i n t e r p o l a t i o n on the
quaternion unit sphere. A l t h o u g h quaternions are less
familiar, conversion to quaternions and generation of
in-between frames can be completely a u t o m a t i c , no
m a t t e r how key frames were originally specified, so
users d o n ' t need to k n o w - - o r c a r e - - a b o u t inner details.
The s a m e cannot be said for Euier's angles, which are
more difficult to use.
Spherical i n t e r p o l a t i o n itself can be used for purposes
besides a n i m a t i n g rotations. For example, the set of all
possible directions in space forms a sphere, the so-called
G a u s s i a n sphere, on which one might w a n t to control
the positions of infinitely d i s t a n t light sources.
Modelling features on a globe is a n o t h e r possible
application.
It is simple to use a n d to p r o g r a m the method proposed
here. i t is more difficult to follow its development.
This stems from two causes: 1) rotations in space are
more confusing t h a n one might think, a n d 2)
i n t e r p o l a t i n g on a sphere is trickier t h a n i n t e r p o l a t i n g
in, say, a plane. R e a d e r s well a c q u a i n t e d with splines
and t h e i r use in c o m p u t e r a n i m a t i o n should have little
difficulty, although e~en t h e y m a y stumble a bit over
quaternions.
2. D e s c r i b i n g
rotations
2.1 Rigid m o t i o n
Imagine hurling a brick t o w a r d s a plate glass window.
As the brick flies closer and closer, a n e a r b y physicist
t
Author's current address: 1700 Santa Cruz Ave., Menlo Park,
CA 94025
245
@ S I G G R A P H '85
might observe t h a t , while it does not change shape or
size, it can t u m b l e freely. L e o n h a r d Euler proved two
centuries ago t h a t , however the brick tumbles, each
position can be achieved by a single r o t a t i o n from a
reference position. [Euler,1752] [Goldstein] The same is
true for any rigid body. ( S h a t t e r i n g glass is obviously
not a single rigid body.)
T h a t is, the change m a t r i x M is orthogonal; its columns
(and rows) are m u t u a l l y p e r p e n d i c u l a r unit m a g n i t u d e
vectors. Because M m u s t also preserve d e t e r m i n a n t s , it
is a special orthogonal m a t r i x , satisfying
W h i l e t r a n s l a t i o n s are well a n i m a t e d by using vectors,
r o t a t i o n a n i m a t i o n can be improved b y using the
progenitor of vectors, quaternions. Q u a t e r n i o n s were
discovered b y Sir W i l l i a m R o w a n H a m i l t o n in O c t o b e r
of 1843.
The m o m e n t is well recorded, for he
considered t h e m his most i m p o r t a n t contribution, the
inspired answer to a fifteen-year search for a successor
to complex numbers. [Hamilton] By an odd quirk of
m a t h e m a t i c s , only systems of two, four, or eight
c o m p o n e n t s will m u l t i p l y as H a m i l t o n desired; triples
h a d been his s t u m b l i n g block.
It is well known, a n d a n y h o w easy to show, t h a t the
special o r t h o g o n a l m a t r i c e s form a group, SO(3), u n d e r
m u l t i p l i c a t i o n . [MacLane][Goldstein][Misner] In this
r o t a t i o n group, the inverse of M is j u s t M t, the
opposite r o t a t i o n .
Soon a f t e r quaternions were introduced, A r t h u r Cayley
published a w a y to describe r o t a t i o n s using the new
m u l t i p l i c a t i o n . [Cayley] The n o t a t i o n in his p a p e r so
closely a n t i c i p a t e s m a t r i x notation, which he devised
several y e a r s later, t h a t it m a y be t a k e n as a formula
for converting a quaternion to a r o t a t i o n matrix. It
t u r n s out t h a t the four values m a k i n g up a quaternion
describe r o t a t i o n in a n a t u r a l way: three of t h e m give
the coordinates for the axis of r o t a t i o n , while the
f o u r t h is d e t e r m i n e d by t h e angle r o t a t e d through.
[Courant & Hilbert]
Since c o m p u t e r graphics leans
o p e r a t i o n s , it is p e r h a p s easiest to
and r o t a t i o n m a t r i c e s in terms
history. However quaternions can
as an elegant a l g e b r a of space.
[MacLane]
2.2 R o t a t i o n
heavily on v e c t o r
explain quaternlons
of these, reversing
s t a n d on their own
[Herstein] [Pickert]
matrices
T h a t a t u m b l i n g brick does not change size, shape, nor
"handedness" is m a t h e m a t i c a l l y expressed as the
p r e s e r v a t i o n of dot products a n d cross products, since
these measure lengths, angles, and handedness. A n d
since the d e t e r m i n a n t of a 3X3 m a t r i x can be c o m p u t e d
as the dot p r o d u c t of one column w i t h the cross
p r o d u c t of the o t h e r two, d e t e r m i n a n t s are also
preserved. Symbolically:
Rot(~l)'Rot(-u2) = ~i'~2
Rot(-/11)XRot(~2) = Rot(.121)O22)
det(Rot(_ul),Rot(~2),Rot(.u3) ) ----det(.ul,.U2,~3)
A n i m m e d i a t e consequence is t h a t o r i e n t a t i o n changes
m u s t be linear o p e r a t i o n s , since the preserved
operations
are;
hence
they
have
a
matrix
r e p r e s e n t a t i o n , M . Using the m a t r i x form of a dot
p r o d u c t , 32~ a22, we can s a y more precisely t h a t
( M a21)t ( M 322) = 32~ J22, from which it follows t h a t
M t M=I.
246
d e t ( M ) = +1
To illustrate, the m a t r i x
M =
1
0
0
0 cos ~ --sin
0 sin ~
]
cos
effects a r o t a t i o n t h r o u g h an angle of ~9 a r o u n d t h e x
axis. A f t e r verifying the properties discussed so far,
note t h a t the diagonal entries sum to l + 2 c o s 0. While
it is too lengthy to show here, the diagonal sum
m e a s u r e s the same q u a n t i t y for m a t r i c e s g e n e r a t i n g
r o t a t i o n a r o u n d a n y axis. [MaeLane]
2.3
Quaternions
Quaternions,
like rotations,
also form
a nonc o m m u t a t i v e group u n d e r their multiplication, a n d
these two groups are closely related.
[Goldstein]
[Pickert][Misner] In fact, we can s u b s t i t u t e quaternion
m u l t i p l i c a t i o n for r o t a t i o n m a t r i x multiplication, a n d
do less c o m p u t i n g as a result. [Taylor]
To perform quaternion a r i t h m e t i c , group the four
c o m p o n e n t s into a real p a r t - - a scalar, a n d an
i m a g i n a r y p a r t - - a vector. A d d i t i o n is easy: a d d s c a l a r
to s c a l a r a n d v e c t o r to vector. But our m a j o r i n t e r e s t
is in m u l t i p l i c a t i o n . S t a r t with a simple case: m u l t i p l y
two quaternions w i t h o u t real p a r t s , or more precisely,
with zero real p a r t s . T h e result quaternion has a
vector t h a t is the cross p r o d u c t of the two vector p a r t s ,
and a s c a l a r t h a t is t h e i r dot p r o d u c t , negated:
• 1~2 = [ ( - a l " ~ ) , ( ~ l x - ~ ) ]
It is c e r t a i n l y convenient to e h e o m p a s s b o t h v e c t o r
p r o d u c t s with a single quaternion p r o d u c t . (One early
lover of quaternion a l g e b r a called v e c t o r a l g e b r a a
" h e r m a p h r o d i t e monster", since it required two kinds of
p r o d u c t , each yielding a different t y p e of result.) If one
q u a t e r n i o n has only a s c a l a r p a r t , with its v e c t o r
c o m p o n e n t s all zero, multiplication is j u s t real
m u l t i p l i c a t i o n a n d vector scaling. Combining the two
effects gives the general rule [Brady]:
[81,~1] ['52,.-/22]= [(8182--121".122),(81.122+82jj.1-Jr-.illX222)].
SAN FRANCISCO JULY 22-26
E x c e p t for the cross p r o d u c t this looks like complex
multiplication, (al+ibl)(a2+ib2)
---- (ala2--blb2) +
i(a lb2Wa2bl), as H a m i l t o n intended, t
Q u a t e r n i o n s m u l t i p l y with a cross product because
r o t a t i o n s confound axes. To illustrate , place a book in
front of you, face up, w i t h the top f a r t h e s t away. Use
this orientatiori as a reference. Now hold the sides and
flip it t o w a r d you onto its face, r o t a t i n g 180 degrees
a r o u n d a left-to-right axis, y. Then, keeping it face
down, spin it clockwise 180 degrees a r o u n d an up-down
z axis. Two rotations a r o u n d two p e r p e n d i c u l a r axes;
yet the t o t a l change in o r i e n t a t i o n must be, according
to Euler, a single rotation. Indeed, if you hold the ends
of the spine a n d flip the book 180 degrees a r o u n d this
third, o u t w a r d - p o i n t i n g , x axis, you should restore the
original
orientation.
As
quaternions,
this
is
- - a n t i c i p a t i n g developments a h e a d - - [0,(0,1,0)] times
[0,(0,0,1)] equals [0,(1,0,0)]; the cross product is
essential.
Notice how quaternion o p e r a t i o n s give a new
orientation, in "quaternion coordinates", much as
t r a n s l a t i o n s give a position, relative to some s t a r t i n g
reference. A central message of this p a p e r is t h a t
quaternlon coordinates are best for interpolating
orientations. F o r comparison, imagine using spherical
coordinates for translations! Q u a t e r n i o n s represent
o r i e n t a t i o n as a single rotation, j u s t as r e c t a n g u l a r
coordinates represent position as a single vector.
T r a n s l a t i o n s combine by adding vectors; rotations, by
multiplying quaternions.
The s e p a r a t e
axes of
t r a n s l a t i o n s d o n ' t i n t e r a c t ; the axes of r o t a t i o n s must.
Q u a t e r n i o n s preserve this interdependence n a t u r a l l y ;
Euler's angle coordinates ignore it.
2.4 E u l e r ' s angles
W h y , then, do so m a n y a n i m a t o r s use Euler's angles?
Mostly, I suspect, because quaternions are unfamiliar.
Unlike Euler's angles, quaternions are not t a u g h t early
in s t a n d a r d m a t h a n d physics curricula. C e r t a i n l y
there is a p l e t h o r a of a r g u m e n t s a g a i n s t angle
coordinates.
Euler's
angle
coordinates
specify
orientation as a series of three independent, rotations
about pre-chosen axes. For example, the orientation of
an airplane is sometimes given as "yaw" (or "heading")
around a vertical axis, followed by "pitch" around a
horizontal axis through the wings, followed by "roll"
around the nose-to-tail line. These three angles must
be used in exactly the order given because rotations do
not commute. The ordering of rotation axes used is a
matter of convention, as is the particular set of axes,
no matter what the order. For instance some physicists
use the body centered axes z-x-z, in contrast to the
aeronautics
z-y-x.
At
least a dozen
different
conventions are possible for which series of axes to use.
[Kane][Goldstein] The geometry of orientations in
Euler's angle coordinates is contorted, a n d varies with
choice of initial coordinate axes.
There is no
Hamilton wrote a quaternion as s+iv~÷jv~+kv~, with i 2 =
jq = k 2 = i j k ------1. The multiplication rules given before are
consequences of this elegant formulation.
Volume 19, Number 3, 1985
reasonable w a y to "multiply" or otherwise combine two
rotations. Even converting between r o t a t i o n m a t r i c e s
a n d angle coordinates is difficult a n d expensive,
involving a r b i t r a r y a s s u m p t i o n s a n d trigonometric
functions. In their defense, it must be said t h a t t h e y
are h a n d y for solving differential e q u a t i o n s - - w h i c h is
how Euler used them. [Euler,1758]
3. I n - b e t w e e n i n g a l t e r n a t i v e s
3.1 S t r a i g h t line i n - b e t w e e n i n g
It is not i m m e d i a t e l y obvious how to in-between even
two r o t a t i o n keys. W h a t o r i e n t a t i o n s should an o b j e c t
assume on its j o u r n e y between them? A n a t u r a l answer
is: t a k e the first key as a reference, a n d represent the
second b y describing the single r o t a t i o n t h a t t a k e s you
to it, according to Euler's theorem. The in-between
o r i e n t a t i o n s should be positioned along t h a t r o t a t i o n .
If we plot quaternions as points in four-dimensional
space, the s t r a i g h t lines between t h e m give o r i e n t a t i o n s
i n t e r p o l a t i n g the end points in exactly the above sense.
If we plot Euler's angle coordinates instead, the inbetween orientations will t r y to twist a r o u n d three
different axes simultaneously. This angle i n t e r p o l a t i o n
t r e a t s the three angles of r o t a t i o n at each key
o r i e n t a t i o n as a three-dimensional vector whose
components are i n t e r p o l a t e d i n d e p e n d e n t l y from key to
key. P a r a d o x i c a l l y , we can not r o t a t e simply except
a r o u n d the special axes chosen for composition. W e
m a y even encounter so-called "gimbal lock", the loss of
one degree of r o t a t i o n a l freedom. G i m b a l lock results
from trying to ignore the cross p r o d u c t i n t e r a c t i o n of
rotations, which can align two of the three axes.
Q u a t e r n i o n s are safe from gimbal lock, a n d so have
been used for y e a r s to handle s p a c e c r a f t , where it is
unacceptable. [Kane][Mitchell]
3.2 H o w q u a t e r n i o n s r o t a t e
S t r a i g h t lines between quaternions, however, ignore
some of the n a t u r a l g e o m e t r y of r o t a t i o n space. If our
i n t e r p o l a t e d points were evenly s p a c e d along a line, the
a n i m a t e d r o t a t i o n would speed up in the middle. To
see why, we must look a t how a quaternion converts to
a r o t a t i o n matrix. We r o t a t e a vector b y a quaternion
so: m u l t i p l y it on the right by the quaternion and on
the left by the inverse of the quaternion, t r e a t i n g the
vector as [0,~].
vr = Rot(v)
=
q--I 32
q
Though it is not obvious, the result will always be a
vector, with a zero s c a l a r component. Notice how this
guarantees
R o t ( v l ) Rot(v2) = Rot(.ul V2)
which implies t h a t dot and cross products
preserved, e m b e d d e d in t h e q u a t e r n i o n p r o d u c t .
are
The inverse of a quaternlon is o b t a i n e d by negating its
247
@ S I G G R A P H '85
I IIII
vector p a r t and dividing both parts by the magnitude
squared. F o r q ~ [s,_~],
q-1
1
Itlqll 2
;
while from the 4-D geometry comes'~
Slerp(ql,q2;u) _-- sin ( 1 - - u ) ~
sin0
ql+
[ [ q I [2 = s2"4".IZ'.IZ .
Because all effects of magnitude are divided out,, any
scalar multiple of a quaternlon gives the same rotation.
(This kind of behavior is not unknown in computer
graphics; any scalar multiple of a point in homogeneous
coordinates gives the same non-homogeneous point.)
If the scalar p a r t has value w, and the vector p a r t
values x, y, and z, the corresponding matrix can be
worked out to be
sin u_.~
sin0
q2'
where ql'q2 ~ cos 0. The first is simpler for analysis,
while the second is more pra.ctical for applications.
But a n i m a t i o n s t y p i c a l l y have more t h a n two key poses
to connect, and here even our spherical elaboration of
simple linear interpolation shows flaws.
While
orientation changes seamlessly, the direction of r o t a t i o n
changes a b r u p t l y . In m a t h e m a t i c a l terms, we w a n t
higher order continuity. There are lots of ways to
achieve it---off the sphere; u n f o r t u n a t e l y we've learned
too much.
3.4 Rotation geometry and topology
M =
1--2y2--2z 2
2xy+2wz
2xz--2wy
2xy--2wz
1--2x2--2z 2
2yz++2wx
2xz.+2wy
2yz--2wx
1--2x2--2y 2
when the magnitude w2-l-x2-4-y2.+z2 equals 1. The
m a g n i t u d e restriction implies t h a t , plotted in fourdimensional space, these quaternions lie on a sphere of
radius one. Deeper investigation shows t h a t such unit
quaternions carry the amount of rotation in w, as
cos 0/2, while the vector p a r t points along the r o t a t i o n
axis with magnitude sin 0/2. The axis of a rotation is
t h a t line in space which remains unmoved; but notice
t h a t ' s exactly w h a t happens when scalar multiples of .u
are r o t a t e d by [s,~]. Because the cross product drops
out, multiplication commutes, q - ! meets q, m u t u a l
annihilation occurs, a n d the vector emerges unscathed.
Summing the m a t r i x diagonal leads to the formula
s t a t e d for w. The sum equals 4w2~1, but must also be
l + 2 c o s 0. A trig identity, cos 2t9 = 2cos 2 0--1, finishes
the demonstation.
3.3 G r e a t a r c in-betweening
This sphere of unit quaternions forms a sub-group, S 3,
of the quaternion group. F u r t h e r m o r e , the spherical
metric of S 3 is the same as the angular metric of
SO(3). [Misner] F r o m this it follows t h a t we can rotate
without speeding up by interpolating on the sphere.
Simply plot the two given orientations on the sphere
and draw the great circle arc between them. T h a t arc
is the curve where the sphere intersects a plane through
the two points and the origin. We sped up before
because we were cutting across instead of following the
arc; otherwise the p a t h s of rotation are the same.
A formula for spherical linear interpolation from ql to
q2, with p a r a m e t e r u moving from 0 to 1, can be
o b t a i n e d two different ways. F r o m the group structure
we find
Slerp(ql,q2;u ) = q l ( q ~ l q 2 ) ~ ;
No m a t t e r w h a t we do in general quaternion space, the
u l t i m a t e effect must be i n t e r p r e t e d via the sphere; so
we h a d best work there in spite of the difficulty. It is
i m p o r t a n t to grasp this point. The metric structure,
hence the intrinsic geometry, of the rotation group
SO(3) is t h a t of a sphere. Over small regions, meaning
in this case small rotation angles, a sphere looks as if it
is flat. But if we go far enough along a "straight line",
we end up back where we s t a r t e d . W h a t could be more
evident a b o u t rotations? Their very essence is moving
in circles.
Looking b a c k to the book-turning
experiment, the confounding of axes is like traveling on
a sphere: if we go in some direction to a quarter of the
way a r o u n d the sphere, turn 90 degrees, travel the
same distance, then t u r n and travel again, we will
arrive b a c k home, coming in at right angles to the
direction we h e a d e d out. Even more revealing, we can
leave the north pole in a n y direction and end up at the
south pole, j u s t as we can r o t a t e 360 degrees around
a n y axis and end up oriented the same way.
Local geometry does not, however, determine global
topology. C o n t r a d i c t o r y though it m a y seem, the
geometry curves like a sphere, but the topology says
north and south poles are the same! In fact, each pair
of opposite points represents the same rotation. The
reader m a y preserve s a n i t y through two expedients.
One is to see t h a t this, like homogeneous coordinates, is
geometry under perspective projection. The second is
to
restore
spherical
topology • by
including
"entanglements". Physically, t a k i n g an object with
strings a t t a c h e d and rotating it 360 degrees leaves the
strings tangled; y e t - - m o s t o d d - - r o t a t i n g 720 degrees
does not. [Misner][Gardner]
Accepting the topological o d d i t y is more useful here,
but it leaves a minor inconvenience. Namely, when
converting an o r i e n t a t i o n in some foreign form, such as
a matrix, to a quaternion form, which quaternion
should we choose? W h i c h side of the sphere? A n
answer t h a t works well is this. Construct a string of
quuternions through which to interpolate by choosing
t
248
Glenn Davis suggested this formula.
SAN FRANCISCO JULY 22-26
Volume 19, Number 3, 1985
each a d d e d quaternion on the side closest to the one
before. T h e n small changes in o r i e n t a t i o n will yield
small d i s p l a c e m e n t s on the sphere.
\
points
at
o r d e r continuity, p r o b a b l y enough for us. As the curve
goes t h r o u g h its end points it is t a n g e n t to its end
segments. Line up the segments across a join, m a t c h
t h e i r lengths, a n d the curves will piece t o g e t h e r
smoothly. If the key o r i e n t a t i o n s are placed at joints,
t h e n each short curve moves us from one key to the
next, because each piece passes t h r o u g h its ends.
Now, a l t h o u g h the two segments a b u t t i n g a curve
j u n c t i o n should m a t c h each other, one of the segments
can be chosen freely. These choices determine the axis
and speed of r o t a t i o n as we pass t h r o u g h the keys.
The b u r d e n of choice can be passed to the a n i m a t o r of
course, but a u t o m a t i o n is feasible, a n d generally
preferable.
4.2 C h o o s i n g j o i n t s e g m e n t s
Rep,-eSenti~
a proje~ive pJa~e
3.5 Splines
W e are left with the problem of constructing smooth
curves on spheres. About a hundred years after
quaternions appeared, Isaac Schoenberg published a
two p a r t a t t a c k on ballistics a n d a c t u a r i a l problems,
using w h a t he called splines. [Schoenberg] N a m e d by
a n a l o g y to a d r a f t m a n ' s tool, these are i n t e r p o l a t i n g
curves c o n s t r u c t e d from cubic polynomial pieces, with
second order c o n t i n u i t y between pieces. Cubic spllnes
solve an integral equation which says to minimize the
t o t a l "wiggle" of the curve, as measured by the second
derivative. These i n t e r p o l a n t s are very popular, and
the equation can be a u g m e n t e d with L a g r a n g e
multipliers to constrain t h e solution curves t o lie on a
sphere [Courant h: Hilbert]; y e t there are problems.
F i r s t , the a u g m e n t e d equation is much more difficult
a n d expensive t o solve. Second, the curve must a d j u s t
everywhere if one of the points changes; t h a t is, we
have no local control.
3.6 B~zier curves
While Schoenberg invented splines based on numerical
analysis, Pierre Bdzier invented a class of curves, now
called by his name, based on geometrical ideas. In fact,
he showed how to find points on such a curve by
drawing lines and splitting t h e m in regular proportions.
[Bdzier] This is e x a c t l y w h a t is needed. We a l r e a d y
know how to do the e q u i v a l e n t - - d r a w g r e a t arcs and
proportions of arcs---on a sphere. A complete solution
needs only a little more.
4. Spherical B~zier curves
Spherical linear i n t e r p o l a t i o n gives two conflicting arc
segments a t a joint, one on each side. Smooth the
difference with a n even compromise, aiming for a point
h a l f w a y between where the incoming segment would
proceed, a n d where the outgoing segment must a r r i v e . t
a~
Co~$tt'u~ft~ g pot~t for f a ~ e r ~ t
Given successive key quaternions q n - 1 ,
qn,
qn+l
i n t e r p r e t t e d as 4-D unit vectors, the c o m p u t a t i o n for a
segment point a n after q~ is
a n = Biseet(Donble(qn_l,%),q,+])
'
where
D o u b l e ( p , q ) ~. 2(p-q)q
Bisect(p,q) =
-- p
;
t tPWq
,,p+qlZ
The m a t c h i n g point for the segment before qn should
he
4.1 Joining curves
Bgzler curves go t h r o u g h only t h e i r first and last
defining points, b u t we w a n t to i n t e r p o l a t e all our
o r i e n t a t i o n s . The t r i c k is to splice t o g e t h e r s h o r t
Bdzier curves in the m a n n e r of splines. Their c r e a t o r
showed an easy w a y to do this which g u a r a n t e e s first
bn -~ D°uble(an ,qn)
For the numerically knowledgeable, this construction
approximates the derivative at points of a sampled function by
averaging the central differences of the sample sequence.
[Dahlquist & Bj6rk]
249
~~ S I G G R A P H '85
b~÷j = ~2 (o)
to ensure a smooth join, regardless of how a n is chosen.
a~
%
C=lcvl$tingg a
$pliging B¢2iCr 3e3mentS
lo~¢¢her
B ~ z i e r ct)rve
polnf
recursivel y
5, R e s u l t s
4.3 Evaluating on t h e sphere
5.1 T h e g r a n d s c h e m e
Everything is now in h a n d to imitate B~zier's curve
technique. Each short curve is defined by four
quaternions, qn, an, bn+l, qn+l" Let the p a r a m e t e r u
vary from 0 to 1 as the curve departs qn towards a n
and arrives at qn+l t a n g e n t to the arc from bn+ 1.
Spherically interpolate by proportion u between qn and
an, a n and bn+l, bn+ 1 and qn+l, to obtain three new
quaternions. Then interpolate between those to get
two more; and finally interpolate again, reducing to a
single point. A b b r e v i a t i n g Slerp(p,q;u) as (P:q)=, the
c o m p u t a t i o n looks like this:
qn =p~0)
W h a t have we ended up with? An a n i m a t o r sits at a
workstation and interactively establishes a sequence of
keys for, say, camera orientation. The interpolating
algorithm does not depend on the n a t u r e of the
interface the a n i m a t o r sees; all needed information is
contained in the sequence of keys. P r o b a b l y the
orientations will be represented internally as matrices,
so a conversion step follows. The matrices are "lifted"
to a sequence of neighboring quaternions, qn, on the
unit sphere. Each quaternion within the sequence will
become the endpoint of two spherical B6zier curves.
Between each q u a t e r n i o n pair, qn a n d qn+l, two
(p~O):piO))u=p~ 1)
an =P t °)
(P~') :P i ') )u =p~2)
(ptO):plO))u=pt 1)
bn+l----p~ O)
(p;2):pi2))u=p~3)=qn+u
(p t1) :phi) )u ~-p t2)
qn +1=P t 0)
additional points, a n and bn+l, are added to control
motion through the joints. At this point, time becomes
a p a r a m e t e r along the composite curve. As the frame
n u m b e r increments, the p a r a m e t e r enters and leaves
successive curve pieces. Within each piece a local
version of the p a r a m e t e r is adjusted to run from 0 to 1.
Now the B6zier geometric construction comes into play,
producing a n interpolated quaternion, qn+~, from qn,
an, bn+l, qn+l, and the local parameter, u. F i n a l l y the
mint-fresh interpolated quaternion is t r a n s m u t e d into a
matrix, to be used in rotating a list of object vectors
for rendering.
5.2 Properties
4.4 Tangents
revisited
A simple check proves the curve touches q,~ and qn+l at
its ends. A rather challenging differentiation shows it is
t a n g e n t there to the segments determined by a n and
bn+ 1. However, as with B6zi~r's original curve, the
m a g n i t u d e of the t a n g e n t is three times t h a t of the
segment itself. T h a t is, we are spinning three times
faster t h a n spherical interpolation along the arc.
F o r t u n a t e l y we can correct the speed by merely
t r u n c a t i n g the end segments to one third their original
length, so t h a t a n is closer to qn and b,+ 1 closer to
qn+l"
250
A look at one special case is revealinG. Suppose all the
points to interpolate are spread along a single arc.
This means they represent different a m o u n t s of rotation
a r o u n d a single axis, in which case quaternion
multiplication
commutes.
Under
these
special
conditions, the formula for the curve sections reduces to
qn+u = qn(l-u)8 a : (1-u)~ bn3~'l]-~)u~ qn+l
us
W h e n this is compared to the s t a n d a r d B6zier
polynomial, pn(1--u) 3 + an3(1--u)2u + bn+13(1--u)u ~ +
q,,+lu 3 , it is a p p a r e n t t h a t addition a n d multiplication
SAN FRANCISCO JULY 2 2 " 2 6
have become m u l t i p l i c a t i o n a n d exponentiation. Of
course, when the points are not on one arc,
c o m m u t a t i v i t y fails, so the formula looks much messier.
In the interesting r e s t r i c t e d case when the points are
s p a c e d evenly a n d consecutively a r o u n d an arc, the
resulting a n i m a t i o n behaves exactly as we would hope:
we get smooth,, c o n s t a n t speed r o t a t i o n a r o u n d the
a p p r o p r i a t e axis. Notice t h a t we can choose a n y axis
for this r o t a t i o n .
This is clearly preferable to
i n t e r p o l a t i o n with Euler's angles, where the coordinate
axes are special. A more subtle p r o p e r t y of all
quaternion i n t e r p o l a t i o n is t h a t
the motion is
i n d e p e n d e n t of coordinate axes. So, for example, if we
design a move, t h e n r o t a t e the coordinate system
a r b i t r a r i l y , the g e o m e t r y of the motion will not change.
Euler i n t e r p o l a n t s , u n f o r t u n a t e l y , will do wildly
different things.
5.3 A p p l i c a b i l i t y
R o t a t i o n s in space are significantly more c o m p l i c a t e d
t h a n r o t a t i o n s in a plane. It is easy to deal with the
latter,
since only one
parameter
is involved.
Q u a t e r n i o n s are o u t of place in a plane. Joint control
in robotics simulations has its own highly specialized
body of techniques; and though quaternions have shown
up in the l i t e r a t u r e , t h e y seem less useful in t h a t
context. [Brady] [Taylor] However, B.K.P. Horn has
used a tessellation of the quaternion unit sphere to
identify the o r i e n t a t i o n of an object from its extended
G a u s s i a n image; a good reference is [Brou]. Non-rigid
motion obviously needs to be h a n d l e d specially. But for
moving a c a m e r a eye-point, a n d for m a n y kinds of
o b j e c t motion, quaternion i n t e r p o l a t i o n has strong
advantages.
5.4 Comparisons and complaints
Cost a d v a n t a g e s are difficult to e s t i m a t e . Converting a
m a t r i x to a quaternion requires only one square root
a n d three divides plus some adds, at worst. Converting
b a c k requires g multiplies and 15 adds. While the
conversions d o n ' t use trigonometric functions, the arc
proportioning
does.
For
comparison,
angle
i n t e r p o l a t i o n requires several trigonometric functions as
well as quite a few multiplies a n d adds to create each
i n t e r p o l a t e d m a t r i x . My experience is t h a t the B6zier
scheme is c o m f o r t a b l y fast enough for design work,
which is the only time speed has m a t t e r e d . (If, for
some application, more speed is essential, non-spherical
quaternion splines will u n d o u b t e d l y be faster t h a n
angle i n t e r p o l a t i o n , while still free of axis bias a n d
glmbal lock.)
These i n t e r p o l a n t s are not perfect, of course. Like all
i n t e r p o l a n t s , t h e y can develop kinks between the
i n t e r p o l a t e d points. There are simple algorithms for
adding new sequence points to o r d i n a r y splines w i t h o u t
altering the original curve [Boehm]; t h e y do not work
for this i n t e r p o l a n t . A n d if these curves can be shown
to s a t i s f y some v a r i a t i o n a l principal, it will be b y
chance. It is useful to do this, because a n y solution to
an integral equation like t h a t for splines a d m i t s
subdivision [Lane et all; m i n i m u m c u r v a t u r e between
Volume 19, Number 3, 1985
end points implies minimum c u r v a t u r e between
i n t e r m e d i a t e points as well. Along these lines, Gabriel
and K a j i y a , m o t i v a t e d b y quaternions, have been
developing a technique to find splines on a r b i t r a r y
R e i m a n n i a n manifolds b y solving differential equations.
[Gabriel & K a j i y a ]
6. Questions
F u t u r e research could answer some interesting p r a c t i c a l
questions. W h a t are these spherical B6zier curves? Is
there some a b s t r a c t c h a r a c t e r i z a t i o n of them? Or is
there
some
related
interpolant
that
is wellcharacterized? In light of the success of the geometric
a d a p t a t i o n a p p r o a c h , it a p p e a r s reasonable to a p p l y
the idea to B-splines, which also have a known
geometric e v a l u a t i o n technique. [Cordon & Riesenfeld]
How do spherical B-splines behave? Is it possible to
add new points to a sequence for either kind of curve
without disturbing it? How? C a n B-.splines be m a d e to
i n t e r p o l a t e , not j u s t a p p r o x i m a t e , with a simple
a d j u s t m e n t of control points? Is there a w a y to
construct a curve p a r a m e t e r i z e d by arc length? This
would be very useful. W h a t is the best w a y to allow
varying intervals between sequence points in p a r a m e t e r
space? A b a n d o n i n g the unit sphere, one could work
with the four-dimensional Euclidean space of a r b i t r a r y
quaternlons. How do s t a n d a r d i n t e r p o l a t i o n m e t h o d s
applied there behave when m a p p e d back to matrices?
Note t h a t we now have little guidance in picking the
inverse image for a matrix, and t h a t cusp-free 1~4 p a t h s
do not always project to cusp-free S 3 paths.
However these questions are answered, quaternion
spline i n t e r p o l a n t s a l r e a d y offer a well-behaved
i m p r o v e m e n t over t r a d i t i o n a l techniques. T h e y are
simple to use, simple to implement, robust, efficient,
consistent, a n d flexible. More research would m a k e
them even more so.
7. Acknowledgments
This work was begun for an a n i m a t i o n system I
designed and i m p l e m e n t e d a t Singer-Link. Several
people there deserve t h a n k s , but I especially t h a n k
Glenn Davis, who befriended me with his good humor
and m a t h e m a t i c a l efforts as I struggled through trying
times.
I prefer not to invent the wheel if I can find the plans;
so I p e s t e r e d Don Venhaus, Brian Bar~ky, Tom Duff,
Lance Williams, and Jim Blinn, whom I t h a n k for their
time, their comments, a n d t h e i r assurances t h a t t h e y
h a d not seen this p a r t i c u l a r wheel roll p a s t PDI,
Berkeley, Lucasfilm, NYIT, or JPL.
T h a n k s to everyone a t Pacific D a t a Images for the
interest a n d e n c o u r a g e m e n t t h a t got me s t a r t e d .
The folks at Ridge C o m p u t e r were generous above and
beyond the call of customer s u p p o r t in letting me use
their Imagen t y p e s e t t i n g system to produce this paper.
Lastly, I t h a n k Nori Hail for commenting on numerous
drafts, and more.
251
Q
References
1.
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and Applications, John Wiley and Sons, London
(1072).
2. BOEHM, WOLFGANG, "Inserting new knots into ]3spline curves," Computer-Aided Design 12(4)pp.
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3. BRADY,MICHAEL, "Trajectory Planning," in Robot
Motion: Planning and Control, ed. Michael Brady,
John M. Hollerbach, Timothy L. Hohnson, Tomas
Lozano-Perez and Matthew T. Mason,The M.IT
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4. BROU, PHILIPPE, "Using the Gaussian Image to
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(Winter 1984).
5. CAYLEY, ARTHUR, "On certain results relating to
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141-145 (February 1845).
6. COURANT, R. AND HILBERT, D., Methods o]
Mathematical Physics, Volume I, Interscience Publishers, Inc., New York (1953).
7. DAttLQUIST, GERMUND AND BJiSRCK, A_KE, Numerical Methods, Prentice-Hall, Inc., Englewood CLiffs,
N.J. (1974). Translated by Ned Anderson.
8. EULER, LEONHARD, "Decouverte d'un nouveau
principe de m~canique (1752)," pp. 81-108 in Opera
omnia, Ser. secunda, v. 5, Orell Ffisli Turici,
Lausannae (1957).
9. EULER, LEONItARD, "Du mouvement de rotation
des corps solides autour d'un axe variable (1758),"
in Opera omnia, Ser. seeunda, v. 8, Orell F/isli
Turici, Lausannae 0"
10. GABRIEL, STEVEN A. AND KAJIYA, JAMES T.,
"Spline Interpolation in Curved Manifolds," ,
(1985). Submitted
11. GARDNER, MARTIN, New Mathematical Diversions
from Scientific American, Fireside, St. Louis, Missouri (1971). Chapter 2
12. GOLDSTEIN,HERBERT, Classical Mechanics, second
edition, Addison-Wesley Publishing Company, Inc.,
Reading, Mass. (1980). Chapter 4 and Appendix
B.
13. GORDON, WILLIAM J. AND RIESENFELD, RICHARD
F., "Bernstein-B~zier methods for the computeraided design of free-form curves and surfaces," J.
A C M 21(2) pp. 293-310 (April 1974).
14. GORDON, WILLIAM J. AND RIESENFELD, RICHARD
F., "B-spline curves and surfaces," in Compute~
Aided Geometric Design, ed. Robert E. Barnhill
and Richard F. Riesenfeld,Academic Press, New
York (1974).
15. HAMILTON, SIR WILLIAM ROWAN, "On quaternions; or on a new system of imaginaries in algebra," Philosophical Magazine xxv pp. 10-13 (July
1844).
16. HERSTEIN, I.N., Topics in Algebra, second edition,
John Wiley and Sons, Inc., New York (1975).
252
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17. KANE, THOMAS R., LIKINS, PETER W. AND LEVINSON, DAVID A., Spacecraft Dynamics, McGraw-Hill,
Inc. (1083).
18. LANE, JEFFREY M., CARPENTER, LOREN C.,
WHITTED, TURNER, .AND BLINN~ JAMES F., "Scan
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1980).
19. MACLANE, SAUNDERS AND BIRKHOFF, GARRETT,
Algebra, second edition, Macmillan Publishing Co.,
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20. MISNER, CHARLES W., THORNE, KIP S., AND
WHEELER, JOHN ARCHIBALD, Gravitation, W.H.
Freeman and Company, San Francisco (1973).
Chapter 41 - - Spinors.
21. MITCHELL, E.E.L. AND ROGERS, A.E., "Quaternion
Parameters in the Simulation of a Spinning Rigid
Body," in Simulation The Dynamic Modeling of
Ideas and Systems with Computers, ed. John
McLeod, P.E., (1968).
22. NEWMAN, WH LIAM M. AND SPROULL, ROBERT F.,
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Chapter 21 - - Curves and surfaces.
23. PICKERT, G. AND STEINER; H.-G., "Chapter 8 - Complex numbers and quaternions," in Fundamentals off Mathematics, Volume I Foundations of
Mathematics: The Real Number System and Algebra, ed. H. Behnke, F. Bachmann, K. Fladt, and
W. Sfiss, (1983). Translated by S.H. Gould.
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in
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2B. SMITH, ALVY RAY, "Spline tutorial notes," Technical Memo No. 77, Computer Graphics Project,
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27. SOSS, W., GERICKE, H., AND BERGER, K.H.,
"Chapter 14 - - Differential geometry of curves and
surfaces," in Fundamentals of Mathematics,
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Translated by S.H. Gould.
28. TAYLOR, RUSSELL H., "Planning and Execution of
Straight Line Manipulator Trajectories," in Robot
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Lozano-Perez and Matthew T. Mason,The MIT
Press (1982).
SAN FRANCISCO JULY 22-26
Volume 19, Number 3,1985
L8 Euler angles to quaternion
Appendix I--Conversions
L1 Quatcrnion to matrix
Using the restriction t h a t w2-t-x2-Fy2-t-z2= 1 for u
quaternion q = [w,(x,y,z)], the formula for the
corresponding matrix is
M
[
=
1--2y2--2z2
2xy+2wz
2xy--2wz
2xz+2wy
1--2x2--2z2
2xz--2wy ]
2yz+2wx ].
2yz--2wx
1--2x2--2y 2]
There are twelve possible axis conventions for Euler
angles. The one used here is roll, pitch, and yaw, as
used in aeronautics. A general rotation is obtained by
first yawing around the z axis by a n angle of ¢, t h e n
pitching around the y axis by 0, and finally rolling
around the x axis by ¢. Using the way quaternion
components describe a rotation, we first o b t a i n a
quaternion for each simple rotation.
qrott = [cos@,(sin-~,O,O)]
If the quaternion does not have unit m~gnitude, a n
additional 4 multiplies and divides, 3 adds, and a
square root will normalize it.
(For the matrix
conversion, the square root can be avoided in favor of
divides if desirable.) Now we can obtain the operation
count for creating the matrix. Most terms of the
entries are a product of two factors, one of which is
doubled. So we proceed as follows. First double x, y,
and z, and form their products with w, x, y, and z.
That will take 3 adds and 9 multiplies. Then form the
sum for each of the 9 entries using 1 add each, plus a n
extra add for each of the 3 diagonal elements, for a
total of 12 adds. Thus 9 multiplies a n d 15 adds suffice
to convert a unit quaternion to a matrix.
L2 Matrix to quaternion
An efficient way to determine quaternion components
w, x, y, z from a matrix is to use linear combinations
of the entries Mra,~. Notice that the diagonal entries
are formed from the squares of the quaternion
components, while off-diagonal entries are the sum of a
symmetric and a skew-symmetric part. Thus linear
combinations of the diagonal entries will isolate squares
of components; sums and differences of opposite offdiagonal entries will isolate products among x, y, a n d z
and products with w. Using off-diagonals risks dividing
by a component that may be zero, or within ¢ (the
machine precision) of zero. However we can avoid t h a t
pitfall, and easily compute all components as follows.
w 2 ~ 1/4
(1
+ M l l + M22 + M33)
Z = (MI2 --
FALSE
w=O
X2 -- --1/2 (M22 + 2~'/33)
Multlplying these together in the right order gives the
desired quaternion q -- qyaw qpitch qroll, with components
=
cos --¢ eos --0 cos --¢ + sin -C-sln--0sin
=
si.-
2
y = cos
2
2
2
2
2
cos --0cos --¢ -- cos -C-sin--0sin--¢
2
2
sin £¢os
2
2
2
2
+ sin -C-cos
2
2
2
2
2
sin--¢
2
z = cos -.f-cos 2-sin-¢ - sin -C-sin £eos £
2
2
2
2
2
2
1.4 Euler angles to matrix
Combining the results of the previous two conversions
gives
M =
cos 8cos¢
cos 0sin¢
--sin 0 /
sin ~bsin8cosC--cos~sin ¢ sin ~bsin0sin¢+cos ¢cos 8 cos 0sin~],
cos Csin 0cos¢q.-sinCsin ¢ cos ~sin 0sin¢--sln ¢cos ¢ cos 0cos ~J
1.5 Matrix to Euler angles
x2>c?
M21 ) / 4t.V
q~aw = [c°s 2~,(O,O,sin 2~)]
where ~b, 0, and ¢ are the angles of roll, pitch, and yaw,
respe ctively.
w2>e?
TRUE
qpitch '~ [cos~,(O,sin~,O)]
TRUE
FALSE
!,'2 = 1/2 (1 -- M ~ )
z ~ M m / 2:~
y2>c?
TRUE
z ~ M~3 / 2y
FALSE
z=l
No more t h a n one square root, three divides, and a few
adds and binary scales are required for any conversion.
While converting a matrix to a unit quaternion only
involves the sign ambiguity of square roots, converting
to Euler angles involves inverse trigonometric functions,
as we con only directly determine the sin's and cos's of
the angles. Some convention, such as principle angles,
must be adopted. However interpolation paths will
vary greatly, depending on choice of angles. Setting
t h a t problem aside, here's a way to extract the sin's
and cos's. Looking at the previous equation, sin0 can
be read off directly as --M13. Use the trigonometric
identity cos 0 = ::t=%~"~"~0 to compute cos @to within
a sign, which is the best we can do. Assuming cos0 is
not zero, o b t a i n the sin's and cos's of the other angles
from
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SIGG
sin 0 =
--Mi3
cosO =
sine = M23/cos
cos~b = M a 3 / c o s / ~
s i n e = Ml~ / c o s
~os¢ = M l l / c o s e
If cos t? is zero, t h e n we must avoid dividing by zero. It
also becomes impossible to distinguish roll from yaw.
A d o p t i n g the convention t h a t the yaw angle ¢ is 0
allows
sin ¢ =
--M32
cos ¢ =
M22
sine
=
cos¢ =
0
1
F r o m these values a two a r g u m e n t t a n -1 will give
angles between --Tr a n d +Tr, or 0 and 275 or some other
conventional range; take your pick. (For a faster
conversion, j u s t compute, say, sin -1 a n d check the sign
of the cosine term with respect to cos6~.) Because of
the u n c e r t a i n t i e s of square roots, inverse trigonometric
functions, and yaw-roll separation, matrix to Euler
angle conversion is inherently very ill-defined.
1.6 Quaternion to Euler angles
Use the most straight-forward approach: convert the
q u a t e r n i o n to a matrix, t h e n the matrix to Euler
angles. Of course it is unnecessary to compute matrix
elements t h a t are never used. This conversion is also
u n a v o i d a b l y ill-defined, as quaternions contain no more
information a b o u t angles t h a n matrices do.
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