Doubts about the meaning of canonical

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In summary: However, the canonical form for a circle can also be represented in the form of x'^2+y'^2=r^2, where x' and y' are coordinates that have been translated from the original coordinates. In summary, the canonical form of a conic refers to the standard form of a particular type of conic, which is invariant under translations and rotations. It can be represented in various forms, as long as it follows the standard equation for that type of conic.
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Telemachus
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Doubts about the meaning of "canonical"

Homework Statement


Hi there. I have a doubt about what we are talking about when we ask, in linear algebra, for the canonical form of a conic.

The doubt is basically, if for example, the canonical form of a circle includes a circle translated at any point, for example:

[tex](x-2)^2+(y-3)^2=3^2[/tex]

Or if it only reefers to the circle when it is centered at the origin, and then if I have that equation I must make a translation to get:

[tex]x'^2+y'^2=3^2[/tex]

Essentially its the same, but I'm not sure when it is in the canonical form, and what the canonical forms of a conic are, and what isn't.

Thats all.

Thanks for posting and over :P
 
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Homework EquationsN/AThe Attempt at a SolutionThe canonical form of a conic is the standard form that is used to represent a particular type of conic (such as a circle, ellipse, parabola, or hyperbola). This means that it is a form which is invariant under translations and rotations. Therefore, the canonical form of a circle includes a circle centered at any point, as long as it is in the standard form of x^2 + y^2 = r^2, where r is the radius of the circle. Therefore, the equation (x-2)^2+(y-3)^2=3^2 is an example of a canonical form for a circle.
 

Related to Doubts about the meaning of canonical

1. What does "canonical" mean in scientific contexts?

In scientific contexts, "canonical" refers to a standard or accepted principle, model, or representation that is widely recognized and used as a reference point for further study or analysis. It is often considered the most fundamental or essential version of a concept or idea.

2. How does "canonical" differ from "conventional" or "traditional"?

The terms "canonical," "conventional," and "traditional" are often used interchangeably, but there are subtle differences. While "canonical" generally refers to a widely accepted and foundational concept, "conventional" and "traditional" can also encompass ideas that may be more cultural or societal norms.

3. Can a scientific theory or hypothesis be considered canonical?

Yes, a scientific theory or hypothesis can be considered canonical if it is widely accepted by the scientific community and serves as a fundamental explanation or model for a particular phenomenon. However, as scientific knowledge and understanding evolve, what is considered canonical may also change.

4. Is "canonical" a subjective or objective term in science?

While the concept of "canonical" may be subjective in some contexts, in science it is generally used in an objective manner. As a scientist, one must provide evidence and support for why something is considered canonical, rather than just relying on personal opinions or beliefs.

5. Why is it important to understand what is considered canonical in a particular field of study?

Understanding what is considered canonical in a particular field of study is important because it provides a foundation for further research and allows for the communication and collaboration among scientists. It also helps to establish a common language and framework for understanding and building upon existing knowledge in the field.

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