Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- General introduction
- 1 Mechanism and the mobility of mechanism
- 2 Overconstraint and the nature of mechanical motion
- 3 Some of the various lines in a moving body
- 4 Enumerative geometry and the powers of infinity
- 5 Rigidity and the instantaneous screw axis
- 6 Irregularity and the freedoms within a joint
- 7 The possibilities in reality for practical joints
- 8 Some elementary aspects of two degrees of freedom
- 9 The linear complex of right lines in a moving body
- 10 Line systems and the dual vectors in mechanics
- 11 Geometrical properties of the linear line systems
- 12 The vector polygons for spatial mechanism
- 13 On the two theorems of three axes
- 14 Some reciprocities across the middle number three
- 15 The generality and the geometry of the cylindroid
- 16 The discovery in a mechanism of a cylindroid
- 17 Action, notion, clearances and backlash
- 18 Singular events in the cycles of motion
- 19 Fundamental relations and some algebraic methods
- 20 The special geometry of some overconstrained loops
- 21 The helitangent lines in a moving body
- 22 The cylindroid in gear technology
- 23 The general and the special screw systems
- Bibliography
- Index of proper names
- Subject Index
4 - Enumerative geometry and the powers of infinity
Published online by Cambridge University Press: 07 September 2010
- Frontmatter
- Dedication
- Contents
- Preface
- General introduction
- 1 Mechanism and the mobility of mechanism
- 2 Overconstraint and the nature of mechanical motion
- 3 Some of the various lines in a moving body
- 4 Enumerative geometry and the powers of infinity
- 5 Rigidity and the instantaneous screw axis
- 6 Irregularity and the freedoms within a joint
- 7 The possibilities in reality for practical joints
- 8 Some elementary aspects of two degrees of freedom
- 9 The linear complex of right lines in a moving body
- 10 Line systems and the dual vectors in mechanics
- 11 Geometrical properties of the linear line systems
- 12 The vector polygons for spatial mechanism
- 13 On the two theorems of three axes
- 14 Some reciprocities across the middle number three
- 15 The generality and the geometry of the cylindroid
- 16 The discovery in a mechanism of a cylindroid
- 17 Action, notion, clearances and backlash
- 18 Singular events in the cycles of motion
- 19 Fundamental relations and some algebraic methods
- 20 The special geometry of some overconstrained loops
- 21 The helitangent lines in a moving body
- 22 The cylindroid in gear technology
- 23 The general and the special screw systems
- Bibliography
- Index of proper names
- Subject Index
Summary
Some first thoughts
01. If we agree (and we do) that no more than three parameters are needed to fix a single point in our Euclidean 3-space, we can argue that there are, all counted, an ∞3 of points. Next we can say (and we do) that, because the number of different radii available for the drawing of a sphere about a point is ∞1, there are, all counted, an ∞4 of spheres. Statements like that are typical of enumerative geometry, a method of mathematics much to be used in this book.
02. A parameter, in this metrical context, is a thing to which we can unambiguously ascribe a single number to correspond exactly with its size. Parameters turn out to be, either (a) a distance along some line either straight or curved which is measured from some origin point in the line, or (b) a magnitude of some angle across some surface either planar or ruled which is measured from some origin line in the surface. There appear to be no other kinds of parameter.
03. When we say that there exists for example an ∞n of some geometrical entity we mean, quite simply, that a minimum number n of parameters is needed to specify a nominated one of the entities from among its family. The family in question is then called an n-parameter family. The spheres in space, accordingly, are a 4-parameter family: any one of the spheres can be nominated by the use of only four parameters.
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- Freedom in Machinery , pp. 60 - 63Publisher: Cambridge University PressPrint publication year: 2007