Ttdfjt

I would like to only show the part of Black circles within the Red circle.

Can I do that with Graphics, Circle and/or some sort of Region Constraint?

step = 2 Degree;
[Alpha] = Range[2 Degree, 80 Degree, step];
x = ConstantArray[0, Length@[Alpha]];
y = Sec@[Alpha];
radius = Tan@[Alpha];
range = 1.01;
Graphics[Red, Circle[0, 0, 1], Black, 
 MapThread[Circle[#1, #2, #3] &, x, y, radius], 
 PlotRange -> -range, range, -range, range]

At the moment, I use ContourPlot with RegionFunction option. However, my plot contains a large number of these circles (The amount of the circles shown here is only a quarter for the current step), which makes ContourPlot approach very slow. Moreover, when I zoom in, quite often I find the circles drawn by ContourPlot are not circular, presumably due to PlotPoints and MaxRecursion etc. I tried to play with these two options, but did not succeed in terms of quality (being a circle) and speed.

Thank you!

You can use the three-argument form of Circle:

Graphics[Red, Circle[0, 0, 1], Black, 
 MapThread[Circle[#1, #2, #3, π + ArcTan[#3], 2 π - ArcTan[#3]] &, 
 x, y, radius], 
 PlotRange -> -range, range, -range, range]

Alternatively, use RegionIntersection with Disk[] to get the needed portions of black circles:

circles = MapThread[Circle[#1, #2, #3] &, x, y, radius];

circles2 = RegionIntersection[Disk[], #] & /@ N[circles];

Graphics[Red, Circle[0, 0, 1], Black, circles2, 
 PlotRange -> -range, range, -range, range]

same picture

Update: An alternative way to hide unwanted portions of circles using FilledCurve:

filledCurve = FilledCurve[Line[Append[#, First @ #]& @ 
 CirclePoints[range Sqrt @2, 4]], 
 Line[Append[#, First @ #]& @ CirclePoints[200]]];

Graphics[Red, Circle[0, 0, 1], Black, circles, 
 EdgeForm[None], White, filledCurve, 
 PlotRange -> -range, range, -range, range]

same picture as above

g = Graphics[Red, Circle[0, 0, 1], Black, 
 MapThread[Circle[#1, #2, #3] &, x, y, radius], 
 PlotRange -> -range, range, -range, range];

Show[g, RegionPlot[x^2 + y^2 > 1, x, -1.2, 1.2, y, -1.2, 1.2, 
 PlotStyle -> White]]