I have two layers, 'river (line)' and 'points' for a set of points laying on a line which is perpendicular to the river (for example red points in the first image).

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Given an interval (5.0 m for example), how can I copy those points along the river (blue sets of points in the next image)? But every set of points must be perpendicular to the related segment of river.

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But I have just the following result using the next script.

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from numpy import arange

L = lambda name: QgsProject.instance().mapLayersByName(name)[0]

point_layer = L("points")
river_layer = L("river")

points = list(point_layer.getFeatures())
river_geom = list(river_layer.getFeatures())[0].geometry()

# Find the nearest point on the river to the points
# The points are laying on the same line perpendicular to the river,
# it's sufficient to use any point.
nearest_point = river_geom.nearestPoint(points[0].geometry())

interval = 5.0
start = river_geom.lineLocatePoint(nearest_point) % interval
end = int(river_geom.length()/interval)*interval + start
for i in arange(start, end + interval, interval):
    p = river_geom.interpolate(i)
    dx = p.asPoint().x() - nearest_point.asPoint().x()
    dy = p.asPoint().y() - nearest_point.asPoint().y()

    feats = []
    for point in points:
        feat = QgsFeature()
        geom = point.geometry()
        geom.translate(dx, dy)



You're very close. Now, you just need to rotate your points in the inner for loop.

To know how many degrees, get these 2 angles:

# Before the for loops
n_p_angle = river_geom.interpolateAngle(river_geom.lineLocatePoint(nearest_point))

# Inside the outer for loop
a = river_geom.interpolateAngle(i) 

Now, make sure you import the module math and after the call to geom.translate() call:

geom.rotate(math.degrees(a-n_p_angle), p.asPoint())

You'll get something like this:

enter image description here

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