die Überschreitung
exceeding / violation
The act of not keeping to a set boundary, limit, rule, or regulation.
Die Überschreitung der Geschwindigkeitsbegrenzung wird mit einem Bußgeld geahndet.
Exceeding the speed limit is punished with a fine.

exceeding / crossing
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exceeding / violation
The act of not keeping to a set boundary, limit, rule, or regulation.
Die Überschreitung der Geschwindigkeitsbegrenzung wird mit einem Bußgeld geahndet.
Exceeding the speed limit is punished with a fine.

'Überschreitung' (exceeding/crossing) is the noun formed from the verb 'überschreiten' (to exceed/cross). The verb literally means 'to step over something' or 'to go beyond it'. This basic idea is present in both meanings of the noun.
Man darf die rote Linie nicht überschreiten.
You must not cross the red line.
The noun 'Überschreitung' (exceeding/crossing) is very often used with a genitive object that indicates what is being exceeded or crossed. Alternatively, the preposition 'von' with the dative is also used.
Die Überschreitung des Budgets führte zu Problemen.
Exceeding the budget led to problems.
When rules are involved, 'Überschreitung' (exceeding/violation) is often combined with abstract nouns such as 'Frist' (deadline), 'Limit' (limit), 'Budget' (budget), or 'Kompetenzen' (authority/responsibilities). The context is often formal or legal.
Eine Überschreitung seiner Kompetenzen wurde ihm vorgeworfen.
He was accused of exceeding his authority.
The physical meaning is mainly used in a geographical context or in mountaineering. Typical objects here are 'Gipfel' (summit), 'Pass' (pass), 'Gebirge' (mountain range), or a 'Grenze' (border).
Die Überschreitung des Passes dauert etwa drei Stunden.
The crossing of the pass takes about three hours.
injury / violation
'die Verletzung' and 'die Überschreitung' can both mean that someone does not follow a rule. 'Verletzung' is especially suitable for rights, laws, duties, or agreements. It often sounds as if something important has been disregarded. 'Überschreitung' is especially suitable for clear boundaries or limits, for example speed, time, or a budget. The words can be confused because both can involve something wrong or forbidden. Ask yourself: Was a right or rule broken, or was a limit exceeded?

Wegen der Überschreitung der Höchstgeschwindigkeit musste mein Vater eine Strafe zahlen.
Because he exceeded the speed limit, my father had to pay a fine.
Die Veröffentlichung der privaten Fotos war eine Verletzung ihrer Rechte.
The publication of the private photos was a violation of her rights.