A clipPath is what you want. You can just clip away everything outside the circle.
<svg xmlns="http://www.w3.org/2000/svg" width="80" height="80" onclick='dispcoord(evt)' viewBox="0 0 80 80">
<defs>
<clipPath id="clip1">
<circle cx="40" cy="40" r="39" fill="black" />
</clipPath>
</defs>
<g class="background" stroke-width="3" fill="transparent" clip-path="url(#clip1)">
<circle cx="40" cy="40" r="39" stroke="black" />
<path d="M 0,40 A 37,37 0 0,1 12,12" stroke-width="11" stroke="black" transform="rotate(112.5, 40,40)" />
<path d="M 0,40 A 37,37 0 0,1 12,12" stroke-width="11" stroke="black" transform="rotate(202.5, 40,40)" />
<path d="M 0,40 A 37,37 0 0,1 12,12" stroke-width="11" stroke="black" transform="rotate(292.5, 40,40)" />
<path d="M 0,40 A 37,37 0 0,1 12,12" stroke-width="11" stroke="black" transform="rotate(22.5, 40,40)" />
<circle cx="40" cy="40" r="44" stroke="transparent" stroke-width="7" />
</g>
</svg>