Okay, I need to create a professional wiki document about the (치역) in mathematics, specifically in geometry. Let me by understanding what the user wants. The structure should include a title, overview, detailed sections, and a references section. The needs to be accurate objective, and have enough depth still be easy to understand.

First, the title straightforward: "치역 as the main heading. the overview section should introduce what range is in the context of functions and geometry. I should mention that it's the set of all output values, contrasting with the domain. Maybe start with a simple definition.

Next, the detailed sections. The user mentioned using H2 and H3. So perhaps split into subsections like "정의 및 기본 개념" (Definition and Basic Concepts), "기하학적 해석" (Geometric Interpretation "관련 개념" (Related Concepts), and "응용 분야" (Applications).

In the definition section, I need to define range, maybe give an example with a function. Also, clarify the difference between range and codomain if necessary. Wait, sometimes people confuse codomain and range, so that's a good point to include. Use an example like f(x) = x² where the codomain might be all real numbers but the range is only non-negative.

For the geometric interpretation, how does range apply to geometric functions? Maybe transformations like translations, rotations. For example, if a function maps points in a plane, the range would be the set of all transformed points. Maybe use a linear function like y = mx + b to show how the range is all real numbers if m isn't zero. Or a quadratic function forming a parabola, so the range is limited.

Related concepts could include domain, codomain, function mapping, image. Explain each briefly. Applications might be in real-life scenarios like physics (e.g., projectile motion range), statistics (data ranges), computer graphics. Also, maybe mention how in geometry, the range can represent possible positions after transformations.

I should ensure that technical terms are explained. For example, when talking about functions, define them if needed, but since this is under math and geometry, the audience might have some background. But still, keep explanations clear.

Use markdown formatting: H1 for title, H2 for main sections, H3 for subsections. Examples can be in bullet points or code blocks if it's equations. Wait, equations are usually written in LaTeX, but the user didn't specify code blocks. Maybe just use italics or LaTeX notation within the text.

Need to check for accuracy. For example, in geometry, functions can be transformations, so the range would be the resulting set of points. Also, ensure that examples are correct. For instance, the range of sine function is [-1,1], which is a common example.

Length should be at least 800 characters. Let me estimate: the overview, a few sections with explanations and examples, references. That should cover it.

Possible structure:

치역

개요

간단한 정의, 도메인과의 차이, 중요성.

정의 및 기본 개념

수학적 정의, 함수와 관련된 설명, 예시 함수들.

기하학적 해석

기하학적 변환에서의 치역, 그래프를 통한 설명.

관련 개념

도메인, 공역, 상, 정의역.

응용 분야

물리, 통계, 컴퓨터 그래픽스 등에서의 활용.

참고 자료

간단한 참고문헌이나 링크.

Now, I need to flesh out each section with accurate content. Make sure that each example is correct and relevant. Avoid any markdown errors, like proper heading syntax. Use lists for examples and related concepts. Maybe tables if comparing domain, codomain, range. But the user didn't specify tables, so maybe bullet points are better.

Also, check for any possible mistakes. For instance, sometimes in older terminology, "range" might refer to codomain, but modern usage distinguishes them. So clarify that in the definition section. Mention that in some contexts, especially older texts, "range" might be used differently, but in current mathematics, it's the image.

In geometry, functions can be mappings, so transformations' ranges are the set of all possible images. For example, a rotation in a plane has a range of all points reachable by rotation. Maybe use specific examples like linear transformations.

In applications, projectile motion's maximum horizontal distance is called range, which is a specific application in physics. Also, in statistics, the range of a data set is the difference between the maximum and minimum, but that's a different context. However, since the user specified geometry, maybe focus more on geometric applications rather than statistical range. Wait, the main category is education, math, geometry, so the focus should be on functions in geometry. Maybe projectile motion is part of applied geometry or physics, but since it's under math, it's acceptable.

I should also ensure that the examples are relevant to geometry. For instance, if discussing a function that maps shapes or points in space, the range would be the transformed positions.

Alright, time to start writing the actual content in Korean, following the structure and ensuring clarity and correctness.