
verlaufend
running
Describes a line, shape, or path that continues in a particular direction or in a particular way.
Die schräg verlaufende Naht gibt dem Kleid ein besonderes Design.
The diagonally running seam gives the dress a special design.
running / gradient / progressing
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running
Describes a line, shape, or path that continues in a particular direction or in a particular way.
Die schräg verlaufende Naht gibt dem Kleid ein besonderes Design.
The diagonally running seam gives the dress a special design.

gradient
Describes colors or patterns that flow into one another smoothly and without a sharp edge.
Der Stoff hat ein schön verlaufendes Farbmuster von Blau zu Grün.
The fabric has a beautiful gradient color pattern from blue to green.

progressing
Refers to the course or development of a process or an illness in a particular way.
Bei einer normal verlaufenden Grippe sollte man nach einer Woche wieder gesund sein.
With a normally progressing flu, you should be healthy again after a week.
'Verlaufend' (running/progressing) is the present participle of the verb 'verlaufen' (to run, proceed, or develop). It is used like an adjective and describes something that is currently in a process of 'Verlaufen' (running/progressing) or is characterized by a course or progression.
Die entlang der Küste verlaufende Straße bietet eine wunderbare Aussicht.
The road running along the coast offers a wonderful view.
The context helps you recognize the meaning: If it is about geometry or paths (line, border), meaning 1 is intended. With colors and patterns, meaning 2 is intended. With illnesses or projects (process, development), meaning 3 is intended.
Die Grenze verläuft quer durch den Wald.
The border runs straight through the forest.
In this meaning, 'verlaufend' (running) is often combined with an adverb that specifies the direction or form. Typical examples are 'gerade verlaufend' (running straight), 'parallel verlaufend' (running parallel), or 'im Bogen verlaufend' (running in an arc).
Zwei parallel verlaufende Linien schneiden sich nie.
Two parallel-running lines never intersect.
When it is about processes or illnesses, 'verlaufend' (progressing) is often evaluated by adverbs. Common combinations are 'normal verlaufend' (progressing normally), 'erfolgreich verlaufend' (proceeding successfully), or 'komplikationslos verlaufend' (progressing without complications).
Nach einer komplikationslos verlaufenden Schwangerschaft kam das Baby gesund zur Welt.
After a pregnancy that progressed without complications, the baby was born healthy.
ongoing / continual
For a path or line, „verlaufend“ describes its spatial course and direction. „Laufend“ does not refer to the course there, but rather to something that is currently happening or happening repeatedly. Use „verlaufend“ for the shape of a route and „laufend“ for an ongoing event.

Der Weg ist am Hang entlang verlaufend eingezeichnet.
The path is marked as running along the slope.
Die laufenden Bauarbeiten behindern den Weg noch mehrere Wochen.
The ongoing construction work is blocking the path for several more weeks.