Angular Displacement

Angular displacement

The angle of rotation from the black ray to the green segment is 60°, from the black ray to the blue segment is 210°, and from the green to the blue segment is 210° − 60° = 150°. A complete rotation about the center point is equal to 1 tr, 360°, or 2π radians.

Angular displacement, often denoted by the symbols θ, ϑ, or φ, is precisely what it sounds like: the measurement of an angle through which a physical body has executed a rotation. It quantifies how much a body has revolved or spun around a designated center or axis of rotation. While seemingly straightforward, this concept underpins much of rotational kinematics. The units employed for expressing this displacement are typically radians, degrees, or turns, each offering a different scale for the same fundamental quantity. What makes angular displacement particularly useful, and occasionally a source of minor inconvenience for those who prefer simple magnitude, is that it can be signed. This sign indicates the sense of the rotation—for instance, distinguishing between a clockwise movement and a counter-clockwise one. Moreover, the absolute value of angular displacement isn't constrained to a single turn; a body can rotate multiple times, accumulating a displacement far greater than 360° or 2π radians.

Other names

rotational displacement, angle of rotation

Common symbols

θ , ϑ , φ

SI unit

radian, degree, turn, etc. (any angular unit)

In SI base units

radian (rad)

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Context

When a body embarks on a journey of rotation around its own axis, the dynamics become inherently more complex than merely tracking a single particle. Unlike simple linear motion, where a particle's position can be described with relative ease, a rotating body presents a challenge: every constituent part of that body, unless it's precisely on the axis of rotation itself, experiences a continually changing velocity and acceleration at any given moment. To simplify this rather messy reality, physicists, in their infinite wisdom, often opt to consider the body as "rigid."

A rigid body is, in essence, a convenient theoretical construct where the distances and relative orientations between all its constituent particles are assumed to remain absolutely constant throughout its entire motion. This means no stretching, no bending, no parts spontaneously flying off – a perfectly unyielding entity. This assumption allows for a significant reduction in the complexity of the mathematical models required to describe its motion. Without this simplification, one would have to account for the deformation of the object, the internal stresses, and the energy dissipated through such changes. Of course, in the cold, hard light of reality, nothing is truly rigid; everything possesses some degree of deformability. However, for most practical applications in classical mechanics, the impact of this deformability is often so minimal as to be entirely negligible. It's a useful fiction, much like many other things we accept to make the universe slightly less overwhelming.

Example

Consider the illustration provided – or perhaps you're on a mobile device, so imagine it above. We observe a particle or perhaps a small, idealized body, labeled P, maintaining a fixed distance, r, from a central origin, O. This particle is engaged in a graceful, counterclockwise circular motion. In such a scenario, attempting to track its position using conventional rectangular coordinates (x, y) quickly becomes cumbersome, as both x and y values would be in a constant state of flux, varying with every tick of the clock.

This is precisely where polar coordinates (r, θ) reveal their elegant utility. For our rotating particle P, the value of the radius r remains steadfastly constant – a testament to its fixed distance from the origin. It is the angular position, θ, that dynamically changes over time, precisely encapsulating the rotational aspect of the motion. As P traces its path along the circumference of the circle, it covers an arc length denoted by s. This arc length, a measure of linear distance along the curve, is intrinsically linked to the angular position through a remarkably simple and fundamental relationship:

$s = r\theta.$

Here, s represents the arc length traversed, r is the constant radius of the circular path, and θ is the angular displacement in radians. This equation highlights the direct proportionality between the linear distance covered and the angle swept out, given a constant radius. It's almost as if the universe occasionally throws us a bone by making some things straightforward.

Definition and units

The precise definition of angular displacement is often articulated with the radian as the preferred unit. And frankly, for good reason. The radian offers an inherently elegant and mathematically convenient relationship between the distance traveled along the circumference of a circle (the circular arc length) and the distance r from the center, which is the radius of that circle. This relationship is succinctly expressed as:

$\theta = \frac{s}{r}\mathrm{rad}$

To illustrate this, imagine a body that completes a full 360° rotation around a circle of radius r. The total distance it travels along the circumference is, as any sentient being with a rudimentary understanding of geometry knows, 2πr. Dividing this by the radius r yields the angular displacement:

$\theta = \frac{2\pi r}{r}$

This simplifies with almost embarrassing ease to:

$\theta = 2\pi$

Therefore, one complete revolution is precisely equivalent to 2π radians. This intrinsic connection is why radians are often considered the "natural" unit for angles in physics and mathematics, simplifying countless formulas that would otherwise be cluttered with conversion factors.

This definition is not some arbitrary suggestion; it is a foundational component of the International System of Quantities (ISQ), rigorously formalized within the international standard ISO 80000-3, specifically addressing concepts of Space and time. Its adoption extends into the widely recognized and utilized International System of Units (SI), cementing its status as a global standard.

As previously noted, angular displacement can carry a sign, which serves to unequivocally indicate the sense of the rotation—whether it's clockwise or counter-clockwise. This is not merely a convention but a critical piece of information for accurately describing the kinematics of rotational motion. Furthermore, the absolute value of this displacement isn't confined to a single turn; a body can, and often does, rotate through multiple full circles, resulting in angular displacements far exceeding 2π radians or 360°.

Within the framework of the ISQ/SI, angular displacement plays a crucial role in defining the number of revolutions, N. This quantity is simply derived by dividing the total angular displacement θ by 2π radians (the angular measure of one full revolution): N = θ/(2π rad). As a ratio of two angular quantities, N is, by definition, a quantity of dimension one—a pure, dimensionless number, devoid of units, which is a rather elegant outcome, if one appreciates such things.

In three dimensions

Figure 1 : Euler's rotation theorem. A great circle transforms to another great circle under rotations, leaving always a diameter of the sphere in its original position. Figure 2 : A rotation represented by an Euler axis and angle.

When we venture beyond the comforting confines of two-dimensional planar rotation and into the more intricate realm of three dimensions, angular displacement transforms into an entity possessing both a distinct direction and a quantifiable magnitude. The direction of this entity specifies the axis of rotation, an axis that, by the grace of Euler's rotation theorem, is guaranteed to exist for any arbitrary rotation of a rigid body about a fixed point. The magnitude, in turn, quantifies the extent of the rotation, typically measured in radians, around this very axis. To consistently determine the direction of this axis, one employs the right-hand rule, a simple yet indispensable convention in physics. This combined entity of an axis and an angle is often referred to as an axis-angle representation.

However, a crucial distinction must be made: despite possessing both direction and magnitude, angular displacement in three dimensions is not a vector in the strict mathematical sense. Why, you ask? Because it fails to adhere to one of the fundamental tenets of vector algebra: the commutative law for addition. This means that if you perform two finite rotations in different orders, the final orientation of the body will generally not be the same. The sequence matters, profoundly. For instance, rotating an object 90 degrees around the x-axis, then 90 degrees around the y-axis, will yield a different final orientation than performing the y-axis rotation first, then the x-axis rotation. This non-commutativity is a stark reminder that the world isn't always as straightforward as we might wish.

Nevertheless, there's a small, fleeting moment of reprieve in this complexity. When dealing with infinitesimal rotations—rotations so vanishingly small that second-order infinitesimals can be conveniently, almost dismissively, discarded—the commutative law appears to hold. In this specific, limited scenario, the order of these tiny rotations no longer significantly impacts the final outcome, allowing for a temporary and rather deceptive simplification.

Rotation matrices

For those who insist on describing the universe with more mathematical rigor, several sophisticated methods exist to describe rotations, each with its own merits and complexities. Prominent among these are rotation matrices and Euler angles, though a deeper dive into charts on SO(3) reveals even more exotic representations.

Given that any specific orientation or frame of reference within three-dimensional space can be precisely characterized by a rotation matrix, it logically follows that the angular "displacement" or change in orientation between any two such frames can also be elegantly described by another rotation matrix. If we consider two distinct rotation matrices, let's call them $A_0$ representing an initial orientation and $A_f$ representing a final orientation, the matrix quantifying the angular displacement between them, $\Delta A$ , can be derived through a simple matrix multiplication:

$\Delta A = A_f A_0^{-1}$

Here, $A_0^{-1}$ denotes the inverse of the initial rotation matrix. When this product is performed, particularly when the difference between the initial and final frames is exceedingly small, the resulting $\Delta A$ matrix will be remarkably close to the identity matrix. This proximity to identity is a direct indicator of a minimal angular displacement, hinting at the concept of infinitesimal rotations. In the ultimate limit, as the difference between the frames approaches zero, we arrive at the concept of an infinitesimal rotation matrix.

Infinitesimal rotation matrices

This section is an excerpt from Infinitesimal rotation matrix.

An infinitesimal rotation matrix, sometimes referred to as a differential rotation matrix, is a mathematical construct employed to represent an infinitely small rotation. It's a pragmatic simplification for when the universe decides to briefly cooperate.

While a conventional rotation matrix is characterized as an orthogonal matrix (meaning $R^{\mathsf{T}} = R^{-1}$ ) and represents an element belonging to the special orthogonal group, denoted as $\mathrm{SO}(n)$ , the differential of a rotation behaves rather differently. It manifests as a skew-symmetric matrix (where $A^{\mathsf{T}} = -A$ ). This skew-symmetric matrix resides within the tangent space of the rotation group, specifically the special orthogonal Lie algebra, denoted as $\mathfrak{so}(n)$ . Crucially, an infinitesimal rotation matrix itself is not a rotation matrix in the traditional sense; it's a representation of the change in rotation.

An infinitesimal rotation matrix typically takes the form:

$I + d\theta \, A,$

where $I$ is the ever-present identity matrix, $d\theta$ represents an infinitesimally small angle (so small it's barely worth acknowledging, yet profoundly important), and $A$ is a skew-symmetric matrix that belongs to $\mathfrak{so}(n)$ . This matrix $A$ dictates the axis of the infinitesimal rotation.

For instance, consider the case where $A = L_x$ , which serves as a basis element of $\mathfrak{so}(3)$ and represents an infinitesimal three-dimensional rotation specifically about the x-axis. In this scenario, $L_x$ takes the form:

$L_x = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{bmatrix},$

And consequently, the infinitesimal rotation matrix about the x-axis becomes:

$I + d\theta L_x = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & -d\theta \\ 0 & d\theta & 1 \end{bmatrix}.$

The computational rules for these infinitesimal rotation matrices are, thankfully, somewhat simplified compared to their finite counterparts. The primary concession is that infinitesimals of the second order—terms involving $(d\theta)^2$ or products of different $d\theta$ values—are simply discarded. This strategic omission allows for a significant simplification of calculations. However, it's vital to recognize that with these relaxed rules, these matrices do not rigorously satisfy all the same properties as ordinary finite rotation matrices under the usual, more stringent treatment of infinitesimals. The most notable consequence of this approximation is that the order in which infinitesimal rotations are applied becomes irrelevant. A brief, almost cosmic, moment of commutativity in an otherwise non-commutative universe.

Angular Displacement