# Mirror image of/in/over/across a line

What is the correct preposition to use in a sentence that describes how a mirror image is reflected through something? This is for mathematical use, and I am in doubt about two scenarios, here shown as example sentences:

• The mirror image of point (2,3) in the origin is (-2,-3).
• The mirror image of point (2,3) over the x-axis is (2,-3).

In the first example, a mathematical entity is being mirrored in a point, whereas it in the second example is being mirrored over a line.

Is `in` and `over` correctly used in these two examples? I wouldn't think so as I feel like guessing. Other suggestions than `in` and `over` might be `across`, `of`, `through` and the like. In my native Danish language I would say `in` for a point (Danish `i`) and `over` for a line (Danish `(hen)over`), thus my choice for the two shown example sentences.

What would correct prepositions be in English?

My personal favorite preposition for this type of expression, though it may be a bit old-fashioned, is "about": e.g., -(x^2) is the reflection/mirror image of x^2 about the X-axis. "Across" works, too, as do the suggestions in swmcdonnell's answer.

• Thank you for the answer. Would you also use "about" in the case of a point? Commented Aug 23, 2022 at 20:33
• Yep, wouldn't make a difference if it's a point or a set of points. It's just kind of weird to refer to a "mirror image" of a point, since there isn't much of an "image" there. Commented Aug 23, 2022 at 21:38
• Alright, thank you very much. Mathematically you can reflect a point about another point as you would reflect about a line through that point which is perpendicular to the line between the two points. Commented Aug 23, 2022 at 21:40

Mirroring indicates some type of image production in a directly-across type of transition. When you mirror in a surface, you change the sign of the dimension perpendicular to the surface. When you mirror in a point (as in your example) you change the sign of all dimensions.

Therefore, you can use several possible words. Each is acceptable and correct and means the same: in, over, on, through, across, and by are some of them. For example, "mirrored by the x-axis" would be correct, as would any of the other words used insted of "by."

"Mirrored of" would not be correct. Though you could say "the mirror image of point A=(x,y) through the y-axis is (x,-y)."

"Mirrored around" would be questionable since mirroring is not "around," because that indicates rotation. Mirroring is not rotation.

• Here are many written contexts where "mirror image" and "through the axis" occur in close proximity. But as you say, we can use a range of prepositions here. There's no such thing as the "correct" preposition. Commented Aug 23, 2022 at 18:49

The prepositions you suggested are used for possession, association and location. They don't apply here because you are comparing the point with something else.

I would use with respect to, in relation to or relative to to show that you're stating one object's location based on the location of the other object.

The mirror image of point (2,3) in relation to the origin is (-2, -3).

The mirror image of point (2,3) with respect to the x-axis is (2, -3).

and

The mirror image of point (2,3) relative to the y-axis is (-2, 3).

• Thank you for the response. If I understand your answer correctly, then I believe I must have been unclear with my question - my apologies for that. I am not referring to the point or line themselves being mirrored. Rather, I am referring to, say, some point being mirrored in a point and over a line. So I wouldn't expect possession to be the relevant characteristic here. I have updated the question with clearer example sentences. Commented Aug 22, 2022 at 20:44
• I created a new answer based on your revised question. Commented Aug 22, 2022 at 21:23
• @Steeven To say that "some point [is] being mirrored in a point" seems to me almost meaningless, and I have some training in both mathematics and physics. I suspect that you are referring to a situation in which the lines of site all cross at a singel point, nd the image (perhspd inverted) is formed at a locuse twice as far from the object as the point is. If this is the intent, I would say the image is reflected through a point" or perhaps "is reflected via a point." Have I grasped your intention correctly? In any case I doi not think "in" can be the correct word here. Commented Aug 22, 2022 at 21:23