Prove that is case peaucellier mechanism point p moves in a straight line
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Today, we simply define a line as a one-dimensional object that extents to infinity in both directions and it is straight, i.e. no wiggles along its length. But what is straightness? It is a hard question because we can picture it, but we simply cannot articulate it.
In Euclid's book Elements, he defined a straight line as "lying evenly between its extreme points" and as having "breadthless width." This definition is pretty useless. What does he mean by "lying evenly"? It tells us nothing about how to describe or construct a straight line.
So what is a straightness anyway? There are a few good answers. For instance, in the Cartesian Coordinates, the graph of is a straight line as shown in Image 1. In addition, the shortest distance between two points on a flat plane is a straight line, a definition we are most familiar with. However, it is important to realize that the definitions of being "shortest" and "straight" will change when you are no longer on flat plane. For example, the shortest distance between two points on a sphere is the the "great circle"as shown in Image 2.
Since we are dealing with plane geometry here, we define straight line as the curve of in Cartesian Coordinates.
For more comprehensive discussion of being straight, you can refer to the book Experiencing Geometry by David W. Henderson.
Take a minute to ponder the question: "How do you produce a straight line?" Well light travels in straight line. Can we make light help us to produce something straight? Sure but does it always travel in straight line? Einstein's theory of relativity has shown (and been verified) that light is bent by gravity and therefore, our assumption that light travels in straight lines does not hold all the time. Well, another simpler method is just to fold a piece of paper and the crease will be a straight line. However, to achieve our ultimate goal (construct a straight line without a straight edge), we need a linkage and that is much more complicated and difficult than folding a piece of paper. The rest of the page revolves around the discussion of straight line linkage's history and its mathematical explanation.
In Euclid's book Elements, he defined a straight line as "lying evenly between its extreme points" and as having "breadthless width." This definition is pretty useless. What does he mean by "lying evenly"? It tells us nothing about how to describe or construct a straight line.
So what is a straightness anyway? There are a few good answers. For instance, in the Cartesian Coordinates, the graph of is a straight line as shown in Image 1. In addition, the shortest distance between two points on a flat plane is a straight line, a definition we are most familiar with. However, it is important to realize that the definitions of being "shortest" and "straight" will change when you are no longer on flat plane. For example, the shortest distance between two points on a sphere is the the "great circle"as shown in Image 2.
Since we are dealing with plane geometry here, we define straight line as the curve of in Cartesian Coordinates.
For more comprehensive discussion of being straight, you can refer to the book Experiencing Geometry by David W. Henderson.
Take a minute to ponder the question: "How do you produce a straight line?" Well light travels in straight line. Can we make light help us to produce something straight? Sure but does it always travel in straight line? Einstein's theory of relativity has shown (and been verified) that light is bent by gravity and therefore, our assumption that light travels in straight lines does not hold all the time. Well, another simpler method is just to fold a piece of paper and the crease will be a straight line. However, to achieve our ultimate goal (construct a straight line without a straight edge), we need a linkage and that is much more complicated and difficult than folding a piece of paper. The rest of the page revolves around the discussion of straight line linkage's history and its mathematical explanation.
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