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I want to express the above red line in a formal English.

A line in the theta direction from (x0,y0).

Is the above expression correct?

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  • Two comments, firstly technical language questions are often a poor fit, as you will tend to get answers from non-specialists
    – James K
    Sep 4, 2021 at 20:55
  • Secondly your diagram is odd as it seems to have the x and y axes meeting at x0,y0 and not at (0,0)
    – James K
    Sep 4, 2021 at 20:56

2 Answers 2

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As James K notes, when you are dealing with lines like this where the angle is an important defining feature it is simplest to use polar notation and refer to it as "A line segment with angle theta and radius r" (leaving off the part about "radius r" if the line extends forever).

If you must use the Cartesian coordinate system things are more awkward. The best way is to refer to the endpoint, whether in specific terms or using trigonometry:

  • A line segment extending from the origin to the point (x1y1).
  • A line segment extending from the origin to the point [l*cos(theta), l*sin(theta)].

But if you are actually referring to a line (or half-line, more correctly) which begins at the origin and does not terminate, rather than a line segment, you may have to fall back on an ungaily description like

  • A line extending from the origin at an angle theta from the x-axis.

Or if you can determine the slope of the line and it is not too ugly (i.e. not irrational) you can refer to it as

  • The line y=mx.

This is a special case of the slope-intercept way of describing an equation where the intercept b is zero, meaning the line goes through the origin.

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  • Thank you so much. In my situation, "A line extending from $X$ at an angle $\theta$ from the $x$-axis" is the best option.
    – Danny_Kim
    Sep 5, 2021 at 15:32
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That is a half-line from the point (x₀, y₀), making an angle of θ with the x-axis.

But if you are doing maths, it is easier to do maths and describe the line using polar coordinates or as a set of numbers on the Argand diagram.

The locus of points satisfying arg( z - x₀-yi ) = θ

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