Calculating curve for a machine

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Discussion Overview

The discussion revolves around calculating the curve for a machine design, specifically focusing on the geometric relationships involving an arc defined by the equation X axis squared + Y axis squared = 75 squared. Participants explore various mathematical approaches to derive the arc and its properties, including the use of ellipses and Pythagorean principles.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Technical explanation

Main Points Raised

  • One participant notes that there will be two points on the x and y axes, with the line joining them having a length of 75, and suggests considering the locus of all possible positions for that line.
  • Another participant proposes extending the line by another 75 in the same direction and mentions that this extension will trace points defined by the equation P(–y, 2 * x).
  • A participant expresses confusion about how to use online resources to achieve the same arc and seeks clarification on the mathematical process.
  • One participant discusses the properties of the ellipse, stating that the semi-major axis is 150 and the semi-minor axis is 75, and encourages finding the general equation of an ellipse in Cartesian coordinates.
  • A participant reflects on the stationary nature of the vertical axis in their design, emphasizing the need for consistent x, y, and z positions and indicating a pivot point for movement.
  • Another participant shares links to resources, including books on linkages and drawing straight lines, suggesting they may be helpful for understanding the mechanics involved.

Areas of Agreement / Disagreement

Participants express various viewpoints and approaches to the problem, with no consensus reached on a single method or solution. Some participants provide suggestions and resources, while others share their uncertainties and challenges in understanding the mathematics involved.

Contextual Notes

Participants mention the need for specific assumptions regarding the angles and proportions involved in the design, as well as the implications of the stationary vertical axis. The discussion includes references to external resources that may not fully address the unique aspects of the problem at hand.

Phillip81
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Mod note: Thread moved from the Precalc Homework section
1. Homework Statement
So I am designing a machine. the issue is I have to calculate like 8 things and the maths is floating in my head but I can't get there.

the requirements are that X axis squared + Y axis Squared = 75 Squared.
then project a line out another 75 from there to get the arc I am referring to. I have included 2 rough attempts at trying to find the arc but no maths involved.
now the vertical line will always be above the horizontal. but the horizontal can go + or - 75

Homework Equations


to equate the points pythagoras... to equate an arc from this... My brain is overloading

The Attempt at a Solution


I got some pics for ya
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2.png
 

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Welcome to pf.

There will be two points, one on the x and one on the y axis.
By Pythagoras, the line that joins those two points has a length of 75.
The angle of that line is not specified so consider the locus of all possible positions for that line.

If you extend that line by another 75 in the same direction, (say at the x end), the extension will trace the points P( –y, 2 * x ).

Google "ellipse mechanism"
https://en.wikipedia.org/wiki/Trammel_of_Archimedes
 
You know that the extension is the same length as the hypotenuse line = 75.
The semi-major axis of the ellipse, a = 75+75 and the semi-minor axis, b = 75.
Find the general equation of an ellipse in Cartesian coordinates, in terms of x, y and a, b. wikipedia?
You can then write the equation of an ellipse from that information alone.
What would happen if the extension was a different proportion of the diagonal?
 
I realized this was in homework... but its not for a course or school. my bad if someone wants to move it I would appreciate it.

I have rethought the idea.now the vertical axis had to be stationary. this is because the home must always be the same in x y and z. Z is not moving so I only have x and y to move. as the pivot point is the middle point I will use the bottom point to move the item backwards and up. now I have a lot of maths to play with
 

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