Egypt has condemned Israel for its "irresponsible" killing of three police officers on the border with Gaza.
Israelis have been accused of firing indiscriminately in the border zone
An Israeli tank opened fire on the Egyptian police after mistaking them for Palestinian militants trying to cross into the Gaza Strip.
Egyptian Foreign Minister Ahmed Aboul Gheit said Cairo "strongly protested and condemned" the shooting and demanded an urgent inquiry.
Israeli Prime Minister Ariel Sharon has apologised to President Hosni Mubarak.
Earlier, Israeli Deputy Defence Minister Zeev Boim said he regretted what he called a "grave incident".
In a statement released by his office, Mr Sharon promised a quick investigation, and pledged to share the results with Egypt.
It is unlikely, however, that many Egyptians will be disposed to believe the Israeli version, says the BBC's Heba Saleh in Cairo.
She adds that there is so much distrust and anger here against the Jewish state that every Israeli action is thought to conceal a conspiracy.
The Egyptian Foreign Ministry issued a strongly-worded statement despite Israel's apology.
It said: "Egypt condemns and strongly protests this regrettable incident.
"We demand that the Israeli authorities conduct an immediate, thorough and comprehensive investigation into the circumstances that led to this incident, and present an explanation.
"The Foreign Minister has asked Egypt's embassy in Tel Aviv to inform the Israeli foreign minister of the Egyptian protest to this irresponsible behaviour."
The BBC's Alan Johnston in Gaza says there has been continual friction on the Gaza-Egyptian border between Israeli troops and Palestinian militants trying to smuggle in weapons.
The Israelis have often been accused of firing indiscriminately in the area, our correspondent adds.
The three border police were killed after the tank unit fired on "suspect silhouettes", Israeli army radio said, which it believed to be planting explosives on the border.
The shooting occurred in an area marked as "the Philadelphi road" on Israeli military maps - a narrow strip of land on the southern edge of the Gaza Strip.
Israel has long argued it will need to hold on to the area, even if the planned disengagement from Gaza goes ahead.
However, last month an aide to Mr Sharon told the BBC Israel would pull out of the Philadelphi strip as well if a security understanding could be reached with Egypt.
Egypt and Israel have been at peace since 1979.