Chapter 1
Planar Kinematic Differential Geometry

Kinematics, a branch of dynamics, deals with displacements, velocities, accelerations, jerks, etc. of a system of bodies, without consideration of the forces that cause them, while kinematic geometry deals with displacements or changes in position of a particle, a lamina, or a rigid body without consideration of time and the way that the displacements are achieved. As a combination of kinematic geometry and differential geometry both in content and approach, kinematic differential geometry describes and studies the geometrical properties of displacements.

There are a number of articles and books on kinematic geometry. Pioneers such as Euler (1765), Savary (1830), Burmester (1876), Ball (1871), Bobillier (1880), and Müller (1892) established the theoretical foundation and developed the classical geometrical and algebraic approaches for studying kinematic geometry in two dimensions some hundred years ago. The classical geometric and algebraic approaches are still in use today. Differential geometry is favored by many researchers studying the geometrical properties of positions of a planar object, changes in its positions, and their relationships. Invariants, independent of coordinate systems, are introduced to describe the geometric properties concisely. Thanks to the moving Frenet frame for describing infinitesimally small variations of successive positions, the positional geometry can be naturally and conveniently connected to the time-independent differential movement of a planar object.

This chapter deals with the kinematic characteristics of a two-dimensional object (a point, a line) in a plane without consideration of time by means of differential geometry. Though abstract, the explanation is judiciously presented step by step for ease of understanding and will be a necessary foundation for studying the kinematic characteristics of a three-dimensional object by means of differential geometry in later chapters.

1.1 Plane Curves

1.1.1 Vector Curve

A plane curve is represented in rectangular coordinates as

where is a parameter. The above equation can be rewritten in the following way by eliminating the parameter :

or in implicit form as

In a fixed coordinate frame , the vector equation of curve can be written as

or

Obviously, both the magnitude and direction of in equation (1.5) vary.

To describe a curve in the vector form, a real vector function, represented by a unit vector with an azimuthal angle with respect to axis , measured counterclockwise, is defined as a vector function of a unit circle (see Fig. 1.1). A plane curve can be denoted by the following vector function:

In the above equation, the magnitude and direction of vector depend on the scalar function and the vector function of a unit circle .

Figure 1.1 Vector function of a unit circle

Another vector function of a unit circle can be obtained by rotating counterclockwise about by p/2 (in Chapters 1 and 2, is the unit vector normal to the paper and directed toward the reader).

The vector function of a unit circle has the following properties:

1. Expansion 1.7
2. Orthogonality

   For a unit orthogonal right-handed coordinate system consisting of , , and , we have the following identities:

   1.8
3. Transformation 1.9
4. Differentiation 1.10

The descriptive form of a curve depends on the chosen parameters and coordinates. A curve may have many descriptive forms, which differ in complexity if the parameters and reference coordinates are chosen differently. Below are three examples.

Example 1.1

A circle with radius and center point C is shown in Fig. 1.2. Write its equation in both vector and parameter forms.

Figure 1.2 A circle

Solution

The parameter equation of a circle in rectangular coordinates can be written as

where are the coordinates of the center of the circle in the reference frame .

Alternatively, the same circle can be represented as a vector function of a unit circle:

Example 1.2

An involute is shown in Figs 1.3 and 1.4. Write its equation in both vector and parameter forms.

Figure 1.3 An involute

Figure 1.4 An involute with a unit circle vector function

Solution

The equation of an involute can be written in three different forms using polar coordinates, rectangular coordinates, and a vector function of a unit circle, where is the radius of the base circle.

1. Polar coordinates: E1-2.1
2. Rectangular coordinates: E1-2.2
3. Vector function of a unit circle: E1-2.3

Example 1.3

A planar four-bar linkage is shown in Fig. 1.5. Write the equation of the coupler curve in both parameter and vector forms.

Figure 1.5 A planar four-bar linkage

Solution

As shown in Fig. 1.5, links and , in a planar four-bar linkage with link lengths , form an inclination angle and with respect to the fixed link. A moving rectangular coordinate system attached to link BC and a fixed coordinate system attached to the fixed link are established. Point in the coupler with polar coordinates can be represented in the coordinate system as

1. The parameter equation of coupler curves

   A coupler curve traced by point can also be expressed in the fixed frame as

   E1-3.2

   A sextic algebraic equation can be deduced for a coupler curve if parameters and are replaced by function in the displacement solution of a four-bar linkage.

2. The vector equation of coupler curves

   Link AB rotates about joint A of the fixed link AD, and link BC rotates about joint B of link AB. Since a circle can be expressed by a vector function of a unit circle, a coupler curve of a four-bar linkage can be written as

   E1-3.3

   A point in link AB traces a circle vector in the fixed frame . A point in coupler link BC produces a circle vector in the reference frame of link AB. The subscripts inside the brackets are independent variables. Here, we deal with the coupler point relative to the coordinate system by the vector function of a unit circle.

Based on the above three examples, we observe that the description of a plane curve in terms of a vector function of a unit circle is simpler than the traditional algebraic equation. Moreover, since a vector function of a unit circle has intrinsic properties, its successive derivatives with respect to the chosen parameters can be conveniently obtained.

Invariants of a curve, independent of the coordinate system used, can be used to simplify the equation of the curve, which is considered a general rule in differential geometry. The arc length of a curve, which is also termed a natural parameter, is an invariant. Other invariants will be introduced in the later of this chapter and other chapters of the book. For equation (1.4), can be replaced by . The differential relationship between and can be written as

Then, the vector equation of curve is expressed in terms of as

It is recognized that . Using the Taylor expansion, curve can be expressed in the neighborhood of point by

where .

1.1.2 Frenet Frame

In a fixed frame, a curve is traced by a point of a moving body. There exists a connection between the point path and the moving body. A frame that moves along the curve can be employed to study the intrinsic geometrical properties of the curve.

Assume that the unit tangent vector of a plane curve is always in the direction of increasing arc length. Adopting the right-handed rule, as in the case of equation (1.8), the unit normal vector of a curve may be defined as . A unit orthogonal right-handed coordinate system may be uniquely established for each point on the curve. This moving Cartesian reference frame is called the Frenet frame, or the moving frame of a plane curve (see Fig. 1.6). The Frenet frame for a plane curve may be defined as

where , an invariant of the curve, is the curvature. Performing a dot product of both sides of the second equation in (1.14) with vector , we obtain

If a vector equation with a general parameter is given, as in equation (1.4) for a plane curve , the unit tangent vector can be expressed as

Utilizing the identity equation , the unit normal vector is obtained as

According to...