Upper half-plane

Key takeaways

Upper half-plane belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Upper half-plane to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Upper half-plane from memory before moving on to harder problems.

Reference excerpt

In mathematics, the upper half-plane, ⁠ H , {\displaystyle {\mathcal {H}},} ⁠ is the set of points ⁠ ( x , y ) {\displaystyle (x,y)} ⁠ in the Cartesian plane with ⁠ y > 0. {\displaystyle y>0.} ⁠ The lower half-plane is the set of points ⁠ ( x , y ) {\displaystyle (x,y)} ⁠ with ⁠ y < 0 {\displaystyle y<0} ⁠ instead. Arbitrarily oriented half-planes can be obtained via a planar rotation. Half-planes are an example of two-dimensional half-space. A half-plane can be split in two quadrants.

Affine geometry The affine transformations of the upper half-plane include

shifts ( x , y ) ↦ ( x + c , y ) {\displaystyle (x,y)\mapsto (x+c,y)} , c ∈ R {\displaystyle c\in \mathbb {R} } , and dilations ( x , y ) ↦ ( λ x , λ y ) {\displaystyle (x,y)\mapsto (\lambda x,\lambda y)} , λ > 0. {\displaystyle \lambda >0.}

Proposition: Let ⁠ A {\displaystyle A} ⁠ and ⁠ B {\displaystyle B} ⁠ be semicircles in the upper half-plane with centers on the boundary. Then there is an affine mapping that takes

A {\displaystyle A} to B {\displaystyle B} .

Proof: First shift the center of ⁠ A {\displaystyle A} ⁠ to ⁠ ( 0 , 0 ) . {\displaystyle (0,0).} ⁠ Then take λ = ( diameter of B ) / ( diameter of A ) {\displaystyle \lambda =({\text{diameter of}}\ B)/({\text{diameter of}}\ A)}

and dilate. Then shift ⁠ ( 0 , 0 ) {\displaystyle (0,0)} ⁠ to the center of ⁠ B . {\displaystyle B.} ⁠

Inversive geometry Definition: Z := { ( cos 2 ⁡ θ , 1 2 sin ⁡ 2 θ ) ∣ 0 < θ < π } {\displaystyle {\mathcal {Z}}:=\left\{\left(\cos ^{2}\theta ,{\tfrac {1}{2}}\sin 2\theta \right)\mid 0<\theta <\pi \right\}} . ⁠ Z {\displaystyle {\mathcal {Z}}} ⁠ can be recognized as the circle of radius ⁠ 1 2 {\displaystyle {\tfrac {1}{2}}} ⁠ centered at ⁠ ( 1 2 , 0 ) , {\displaystyle {\bigl (}{\tfrac {1}{2}},0{\bigr )},} ⁠ and as the polar plot of ρ ( θ ) = cos ⁡ θ . {\displaystyle \rho (\theta )=\cos \theta .}

Proposition: ⁠ ( 0 , 0 ) , {\displaystyle (0,0),} ⁠ ⁠ ρ ( θ ) {\displaystyle \rho (\theta )} ⁠ in ⁠ Z , {\displaystyle {\mathcal {Z}},} ⁠ and ⁠ ( 1 , tan ⁡ θ ) {\displaystyle (1,\tan \theta )} ⁠ are collinear points. In fact, Z {\displaystyle {\mathcal {Z}}} is the inversion of the line { ( 1 , y ) ∣ y > 0 } {\displaystyle {\bigl \{}(1,y)\mid y>0{\bigr \}}} in the unit circle. Indeed, the diagonal from ⁠ ( 0 , 0 ) {\displaystyle (0,0)} ⁠ to ⁠ ( 1 , tan ⁡ θ ) {\displaystyle (1,\tan \theta )} ⁠ has squared length 1 + tan 2 ⁡ θ = sec 2 ⁡ θ {\displaystyle 1+\tan ^{2}\theta =\sec ^{2}\theta } , so that ρ ( θ ) = cos ⁡ θ {\displaystyle \rho (\theta )=\cos \theta } is the reciprocal of that length.

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Upper half-plane

Start with the simplest possible case. Write down what Upper half-plane claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Upper half-plane before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Upper half-plane ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Upper half-plane

In research: Upper half-plane appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Upper half-plane in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Upper half-plane is common in secondary-school and first-year university syllabi. It links to neighbouring topics Complex analysis, Differential geometry, Hyperbolic geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Upper half-plane outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Upper half-plane in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Upper half-plane means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Upper half-plane out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Upper half-plane in simple terms?

In mathematics, the upper half-plane, ⁠ H , {\displaystyle {\mathcal {H}},} ⁠ is the set of points ⁠ ( x , y ) {\displaystyle (x,y)} ⁠ in the Cartesian plane with ⁠ y > 0. {\displaystyle y>0.} ⁠ The lower half-plane is the set of points ⁠ ( x , y ) {\displaystyle (x,y)} ⁠ with ⁠ y < 0 {\displaystyl…

Why does Upper half-plane matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Upper half-plane?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Upper half-plane.