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The principle called Arc of a Motion is an animation principle that refers to what in real physics we know as trajectory or path of motion. From an animation standpoint, it defines the flow of motion in both time and space. Its main purpose is to give the feeling of an organic and natural movement.

This article introduces multiple applicabilities of the Arc of a Motion principle from the very basic motion of a pendulum, to the bouncing ball animation, to the motion flow of a human walk. Once you’ll understand the logic behind a motion flow you’ll then be able to navigate much more easily through the language of animation.

Arc of motion as a nonlinear motion

In general terms, The Arc of Motion means the visual path (trajectory) of an action from one point to another (keyframes).

This path is a curve and it can be subtle or extreme but is never a linear path (line).

In nature, nothing moves linear or at a constant speed, there’s always some kind of variation of speed. We call this “nonlinear motion” and it means a motion in both time and space along a curved or parabolic trajectory.

Just to give a few common examples: a thrown football, a pitched/batted baseball, a cannonball shot by a cannon out, a ball rolling on the edge of a table.

various real life examples of parabolic trajectories

All these cases have the same key principles:

  • Time and space: there’s an object in motion which travels a certain distance in a certain time period.
  • Trajectory as an arch-path: the object’s motion path is a curve or a parabolic path, it’s never a straight line.
  • Motion happens on two axes: The arch-path trajectory means that the object moves in two ways (directions) at the same time (horizontally and vertically).
  • Speed is variable, the object in motion accelerates or decelerates during a certain period of time, it does not move with constant speed.
  • Acceleration and Deceleration. Acceleration is considered to describe an increased change of speed. On the other hand, deceleration is considered to describe a decrease in speed.

Therefore, the arc of motion refers to a nonlinear motion with a variable speed along a curved trajectory.

Motion arcs of a ball on a string (pendulum)

Fundamentals of animation start with the theory and the practice of Pendulum Animation. This simple exercise sets up the keystones of the Arc of a Motion Principle because it simply explains the basic logic and the physics behind it.

A real pendulum consists of a ball (mass) hanging from a string (of a certain length) and fixed at a pivot point. Released under a certain angle, the pendulum will swing back and forth as periodic motion.

By the way, the word “pendulum” comes from the Latin “pendulus”, meaning ‘hanging’.

A real physics lecture on the theory and math behind the pendulum motion you can find it presented by Walter Lewin in his well known and super engaging physics classes“For the love of Physics”. I really encourage you to check out his amazing physics lecture.

arc of motion applied to a pendulum, captures from the Walter Lewin's lectures, For the Love of Physics book

From the animation standpoint, let me give you an insight into the key aspects.

  • Time: the time for one complete cycle (left to right and right to left swing) is called a Period or Cycle.
  • Length or height: The period depends only on the length of the pendulum, it does not depend on the mass of the ball or the angle of rotation (watch the first 20 min of Walter Levin’s lecture). The shorter the height of the pendulum is, the less time it needs until a complete stop.
  • First swing: The first swing is always the most extreme swing and it never gets overshoot by another swing. Unless an external force (push or pull) acts upon the ball.
  • Trajectory: The motion of the pendulum is according to an arch-path trajectory, always the same exact path.
  • Additional forces: Real pendulums depend on the air friction so the amplitude (angle of rotation) of their swings declines more and more until they stop. But then this is very subtle.

Motion arcs of a free-falling ball (bouncing ball)

Absolutely every single animation book presents the theory behind motion in general through the use of simple ball-characters. This is because there’s no need for powerful graphics in order to create the illusion of movement and aliveness. Just a simple circle (2D) or a sphere (3D) will do.

When it comes to the motion path of a bouncing ball, then it’s all in the arcs!!!

arc of motion applied to a bouncing ball, types of balls and their bounciness level

The general framework of a bouncing ball is:

  • Trajectory: Any kind of free-falling ball, regardless of its material, size or flexibility, will bounce along a certain type of arch-type trajectory (parabola).
  • Bounciness: the more elastic the ball is, the more bounces will be needed before it settles. The ball will keep on bouncing until it consumes its energy.
  • Initial bounce: The first bounce-back of a dropped ball always reaches a height lower to its initial drooping-height. This is also the highest accumulation energy the ball has and with each contact, the ball loses its energy due to friction with the contact surface.
  • Elasticity: The more elastic the ball is, the more it will bounce in time. A flexible ball will become squashed with each contact and will simply act like a compressed spring. The ball will push back the ground and will bounce again and again until it loses its energy.

In the real world, there’s also another factor that influences the bounciness and therefore the motion path; that’s the quality and the type of the surface.

  • The harder the surface is (i.e, hard floor) more successive bounces will take before a complete stop.
  • A soft surface (i.e, grass or carpet) will reduce the ball’s energy, and therefore it would take a shorter time for the ball to stop and of course, there’s also a reduced bounciness.

Motion arcs in a human walk cycle

In his book The Animator’s Survival Kit, Richard Willimas goes in great detail in explaining the mechanics of the human walk. He breaks down into pieces the human body and explains the motion within each body-part. For example, what happens with the arms swing? When do the feet pass through the middle position? How are the hips rotations connected to the chest or, is there anything that happens in toes’ motion?

By the way, in medical terms, “arc of motion” means “range of motion” and it refers to the range of motion measured in degrees of a circle, through which a joint can be extended and flexed. However, this is not quite the current focus of this article but it’s good to be aware of this.

From the animation stand-point, we look at human body mechanics interpreted through character animation. Therefore, the main components of a “normal-adult” walk cycle are:

A walk cycle can be defined by a cyclic series of body movements, where the entire body weight shifts from one leg to another. Walking can be also described as a series of losses and recoveries of balance in the human body.

  • Coordination pattern, walking at comfortable speed the arm-to-leg swing ratio is considered to be normal 1:1 (i.e. one arm swing is associated with one leg swing) but at slow speeds this changes to 2∶1.
  • Heel strike, the moment when the heel first strikes the ground.
  • Foot flat, from heel strike to when the full foot is in contact with the ground.
  • Midstance (passing posture), bodyweight is directly over the stance leg.
  • Heel off, the moment the heel of the stance leg leaves the ground.
  • Toe off, when only the toe of the stance leg is in contact with the ground.

This article is too short to discuss the theory behind the animation walk cycles, on the other hand, the key facts to remember are in relation to the arcs of motion principle. Just like with the bouncing ball or with the pendulum animation, arcs in the human walk created the feeling of flow.

  • A walk cycle can be divided into two steps (major phases).
  • Step One or the Stance phase is from right heel contact to right toe-off. The right foot is one of the grounds supporting the body’s weight.
  • Step Two or Swing phase is from right toe-off to the next right heel contact. The right foot is in the air, being advanced forward for the next contact with the ground.
  • Arm-Leg swing: There’s opposite coordination between the arms and legs swing, when the left foot is in front (clockwise motion) the right arm swings in front (counterclockwise motion).
  • CoM: the Body Center of Mass is located in the hips area and it moves in an arc with the highest points in the passing position (when the weight of the body is completely supported on one leg only).
  • Arms: Arms motion is considered to have a Pendulum Swing Motion with the fixed point being the shoulder.
  • Legs: Legs motion is considered to be an Inverted Swing Pendulum with the fixed point being the ankle.

Tips and Tricks

Tip 1. Use motion arcs to plan your animation in space.

Use the Arc of Motion principle to initially plan the animation scenario. It’s simple and efficient to just visualize the entire animation in space (overall trajectory), then, later on, you’ll figure out the perfect timing and rest of the details. For example, a freestyle bouncing ball where the ball jumps over several obstacles in space. The same thing happens in a parkour scene where the “ball” is a character now.

Tip 2. Visualize and analyze your animation as motion trails.

Any animation software that you’re using has a tool or a simple setting that can do this. It may be called “motion trail” or “motion ghost” but its purpose is to visually display the trajectory of a motion based on a specified start and end time (in frames). Plus, you can adjust or edit the trajectory as well.

Tip 3. Clean your motion graphs.

Especially the 3D animation software has powerful Graph Editors that allow you to adjust and edit any motion on any axis of the 3D system. However, do not confuse “motion graphs” with “motion paths”. Generally speaking, make sure to keep the graphs that do correspond to the motion path, just delete the unnecessary extra keys that are “noise” in the motion.

Takeaway

In summary, the goal of this article was to illustrate that the motion along an arch-type trajectory plays a big role in the smoothness and balance of that motion.

By using even a subtle arc of motion your animation will feel organic and natural, not stiff, nor mechanical.

arc of motion as motion trail in space

When a motion feels organic is characterized by two key factors:

  1. The direction of movement. Motion can be in any direction, on two axes at the same time. However, nature has its own way and some things will never change, not even in animation; such as fire burning upwards or a stone falling downwards.
  2. Variation of speed. Speed means a certain distance covered in a certain time, so where there‘s speed there’s also “distance” and “time”. Objects and humans in motion always have variations in speed, there’s always some acceleration and deceleration.

Nearly every movement in nature follows a slightly or an extreme circular trajectory path.

It can be just a subtle head turn from left to right side, a large arm swings from down to up, a ball bouncing all over the place, a frog jumping from one leaf to another. It can be just a leaf falling down from a tree branch. Or, it can be an energetic dancer on a stage. All the motion must feel organic and must have an arch-type trajectory (subtle or extreme).

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Cristina Zoica Dumitru

Cristina Zoica Dumitru

I teach digital art courses online and on-location. My mission is to challenge students to reach their maximum potential for creativity and authenticity.

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